Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations
S\'{e}rgio R. Fenley \thanks{Reseach partially supported by NSF grants DMS-0296139 and DMS-0305313.} \footnote{Mathematics Subject Classification. Primary: 53C23, 57R30, 37D20; Secondary: 57M99, 53C12, 32Q05, 57M50.}
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\begin{document}

\title{Geometry of foliations and flows I: 
Almost transverse pseudo-Anosov flows and 
asymptotic behavior 
of foliations}
\author{S\'{e}rgio R. Fenley
\thanks{Reseach partially supported by NSF grants 
DMS-0296139 and DMS-0305313.}
\footnote{Mathematics Subject Classification.
Primary: 53C23, 57R30, 37D20; Secondary: 57M99, 53C12,
32Q05, 57M50.}
}
\maketitle

\vskip .2in

{\small{
\centerline{\bf {Abstract}}
Let $\fol$ be a foliation in a 
closed  $3$-manifold with negatively curved fundamental group
and suppose that $\fol$ is almost transverse to a
quasigeodesic pseudo-Anosov flow.
We show that the leaves of the foliation
in the universal cover extend continuously to the sphere
at infinity, therefore the limit sets of the leaves are continuous images
of the circle.
One important corollary is that  if
$\fol$ is  a
Reebless, finite depth foliation in a hyperbolic manifold, then
it has the continuous extension property.
Such finite depth foliations 
exist whenever
the second Betti number is non zero.
The result also applies to other classes of foliations, including
a large class of foliations where all leaves
are dense and infinitely many examples with one sided
branching.
%The tools in the proof involve topological properties 
%of pseudo-Anosov flows and transverse/almost transverse
%foliations and also metric properties of flows.
One key tool is a detailed understanding
of asymptotic properties of almost pseudo-Anosov singular
$1$-dimensional foliations in the leaves of $\fol$ lifted
to the universal cover. 
}}


\section{Introduction}

A $2$-dimensional foliation in a $3$-manifold is called Reebless
if it does not have a Reeb component: a foliation of the solid
torus so that the boundary is a leaf and the interior is foliated
by plane leaves spiralling towards the boundary.
As such the boundary leaf does not inject in the fundamental
group level and is compressible. Novikov \cite{No} showed that 
Reebless foliations and the underlying manifolds have excellent
topological properties. This result was extended by Rosenberg
\cite{Ros}, Palmeira \cite{Pa}
and many others.

The goal of this article is to analyse geometric properties of
foliations. Let $\fol$ be a Reebless 
%transversely oriented
foliation in $M^3$ with negatively curved fundamental
group.
Reebless implies that $M$ is irreducible \cite{Ros}.
In this article we will not make use of Perelman's 
fantastic results \cite{Pe1,Pe2,Pe3}, which if confirmed
imply that the manifold is hyperbolic.
Reebless foliations exist for instance  whenever $M$ is irreducible,
orientable and
the second homology of $M$ is not finite \cite{Ga1,Ga3}.
They also exist in much more generality by work of Roberts 
\cite{Ro1,Ro2,Ro3}, Thurston \cite{Th5} and many others.

Let 
$M^3$ be closed, irreducible with negatively curved 
fundamental group.
The universal cover is canonically compactified with a sphere
at infinity (denoted by $\si$), with compactification
a closed ball \cite{Be-Me}. The covering transformations
act by homeomorphisms in the compactified space.
Let $\fn$ be the lifted foliation to the universal cover $\mi$.
The leaves of $\fn$ are topological planes \cite{No}
and they are properly embedded. 
Hence they only limit
in the sphere at infinity. For hyperbolic manifolds,
the relationship between objects
in hyperbolic $3$-space (isometric to $\mi$) and their
limit sets in the sphere at infinity is central to  
the theory of 
such manifolds \cite{Th1,Th2,Mar}. The same is true
if $\pi_1(M)$ is negatively curved \cite{Gr,Gh-Ha}.
There is
a metric in $M$ so that all leaves of $\fol$ are hyperbolic
(that is constant curvature $-1$) \cite{Ca} and so the universal
cover of each leaf of 
$\fol$ is isometric to the hyperbolic  plane ($\hh$).
The {\em continuous extension question} asks whether these
leaves extend continuously to the sphere at infinity,
that is: given the inclusion map from a leaf $F$ of $\fn$
to $\mi$ is there a continuous extension to a map
$F \cup \pin F$ to $\mi \cup \si$? 
Here $\pin F$ is the ideal boundary of $F$ which is homeomorphic
to a circle.
If this is true we say that $\fol$ has the 
{\em continuous extension property}.
In that case the 
restriction of the map to $\pin F$ expresses the limit
set of $F$ as the continuous image of a circle, showing
it is locally connected.





In this article we prove the continuous extension property
for a very large class of foliations.
A pseudo-Anosov flow is {\em almost transverse} to a foliation
if an appropriate blow up of the flow is transverse 
to the foliation.
A blow up is obtained by replacing a (possibly empty) collection
of singular orbits by a union of annuli.
Another property that is important is a metric property:
A flow is {\em quasigeodesic} if it is uniformly
efficient up to a bounded multiplicative
distortion in measuring distances in the universal cover.
This is extremely important for manifolds with
negatively curved fundamental group \cite{Th1,Th3,Gr}.
Our main result is the following:


\vskip .1in
\noindent
{\bf {Main theorem}} $-$ Let $\fol$ be a foliation in
$M^3$ closed, atoroidal.
Suppose that $\fol$ is almost
transverse to a quasigeodesic pseudo-Anosov flow $\Phi$,
which has some prong singularity (that is, not a topological
Anosov flow).
This implies that $M$ has
negatively curved
fundamental group. 
Then $\fol$ has the continuous extension property.
Therefore the limits sets of leaves of $\fn$ are locally 
connected.
The set of foliations almost transverse to a flow
is open in the Hirsch topology of foliations.
\vskip .1in


Notice that the hypothesis imply that $\fol$ is transversely
orientable.
Since $M$ has a singular pseudo-Anosov flow then
$M$ is irreducible and the stable/unstable foliations
of $\Phi$ split to genuine laminations in $M$.
A fundamental result of Gabai and Kazez \cite{Ga-Ka}
then implies that $M$ has negatively curved fundamental
group. For simplicity of statements we usually
use the group negative curvature hypothesis,
but in most places that could be substituted
by the atoroidal hypothesis.

%What is new in this result? The key hypothesis is
%that $\Phi$ is quasigeodesic. Here there are no restrictions
%on the metric behavior of the stable/unstable foliations
%The result also allows for almost transverse flows
%as opposed to transverse flows. This is necessary in
%important applications as the case of finite depth
%foliations shows, see below.

Notice that it is not necessary to assume 
that $\fol$ is
Reebless $-$ we prove that 
the condition of being almost transverse to a pseudo-Anosov
flow implies that $\fol$ is Reebless.

As a first consequence we prove the continuous
extension property for all Reebless finite depth foliations
in hyperbolic $3$-manifolds. 
Roughly a foliation is {\em finite depth} if all leaves
are proper and the leaves are perfectly fitted along
the cutting surfaces of a hierarchy of the manifold.
In particular there are
compact leaves.
These foliations exist whenever the second
homology is not finite.

\vskip .1in
\noindent
{\bf {Corollary A}} $-$ Let $\fol$ be a Reebless finite depth 
foliation in $M^3$ closed hyperbolic. Then $\fol$ has
the continuous extension property. In particular the
limit sets of the leaves are all locally connected.
\vskip .1in



Hence any orientable, hyperbolic $3$-manifold with non finite
second homology has such a foliation with the continuous
extension property. 
Conjecturally any closed, hyperbolic $3$-manifold
has a finite cover with positive first Betti number.
This would
imply there is always a foliation with the continuous extension
property in a finite cover.
The proof of corollary A is simple given previous results:
If necessary take a double cover and assume
that $\fol$ is transversely oriented. Then
Mosher and Gabai proved that such $\fol$ is almost transverse
to a pseudo-Anosov flow $\Phi$ \cite{Mo5}. 
We proved, jointly with Mosher that these
flows 
%almost transverse to finite depth
%foliations 
are quasigeodesic \cite{Fe-Mo}.
The main theorem then implies corollary A.
The result in \cite{Fe-Mo} is only for finite depth foliations: 
the proof depends heavily on the existence of a compact
leaf, $M$ being hyperbolic and the direct association with
a hierarchy.
%Here we assume that $M$ is hyperbolic as opposed to fundamental
%group being negatively curved.
%This is because, given that $\fol$ is
%finite depth, there is a compact leaf, which is an incompressible
%surface. The manifold is also irreducible \cite{Ros} and then
By Thurston's geometrization theorem 
\cite{Th1,Th2,Mor}
it suffices to assume that
$M$ is atoroidal.



\vskip .1in
\noindent
{\bf {Corollary B}} $-$ There are infinitely many 
foliations with all leaves dense which have the continuous
extension property. Many of these have one sided branching.
%These are not uniform foliations.
\vskip .1in


Foliations with all leaves dense can be obtained for example
starting with finite depth foliations and doing small perturbations 
$-$ keeping it still almost transverse to the same 
quasigeodesic pseudo-Anosov flow.
A construction is 
carefully explained by Gabai \cite{Ga3}, providing
infinitely many examples with dense leaves to which 
corollary B applies. 
The examples occur whenever the second Betti number
of $M$ is non zero.
In fact whenever a foliation $\fol$ satisfies the
hypothesis of the main theorem, then any $\fol'$ sufficiently
close to $\fol$ will also be transverse to the same
flow. By the main theorem again, 
$\fol'$  will have the continuous extension
property.
This perturbation feature of the main theorem is not
shared by any previous result
concerning the continuous extension property.
%, see
%the historical remarks section.

A foliation $\fol$ is ${\bf R}$-{\em covered} if 
the leaf space  of $\fn$ is homeomorphic to the
real numbers. Equivalently this leaf space is Hausdorff.
A foliation which is not $\rrrr$-covered has {\em branching},
that is there are non separated points in the leaf space.
This leaf space is oriented (being a simply connected, perhaps
non Hausdorff $1$-manifold) and there is a notion of branching
in the positive or negative directions. If it branches only
in one direction the foliation is said to have 
{\em one sided branching}.
Foliations with one
sided branching, where all leaves are dense and the
foliation is transverse to a suspension
pseudo-Anosov flow (which is quasigeodesic)  were constructed
by Meigniez \cite{Me}. This provides infinitely 
many examples with one sided 
branching to which corollary B applies.


A very important tool in the proof of the main theorem is
an analysis of the topological structure
of the pseudo-Anosov flow. 
Let $\Phi_1$ be the original pseudo-Anosov flow almost transverse
to $\fol$. To make the flow transverse to $\fol$ one
needs in general to blow up a collection of singular orbits
into a collection of flow saturated annuli so that
each boundary is a closed orbit of the new flow $\Phi$.
The blown up flow is called an {\em almost pseudo-Anosov flow}
(see section 3).
If $\wwp$ is the lifted flow to
the universal cover $\mi$ and $\oo$ is its orbit space, then
$\oo$ is homeomorphic to 
the plane $\rrrr^2$ \cite{Fe-Mo} $-$ this is true for
pseudo-Anosov and almost pseudo-Anosov flows. 
When one blows up some singular orbits into a collection of
joined annuli, the stable/unstable singular foliations also
blow up. The two new singular foliations $\Lambda^s, \Lambda^u$
are everywhere transverse
to each other
except at the singularities and the blown up annuli. The blown
up annuli are part of both singular foliations. 
Since
$\fol$ is transverse to the blown up foliations, then the
stable/unstable foliations $\Lambda^s, \Lambda^u$
 induce singular $1$-dimensional foliations
in the leaves of $\fol$ and hence also in the leaves
$\fn$. The behavior of this is
described in the following result, which is
of independent interest also: 

\vskip .1in
\noindent
{\bf {Theorem C}} $-$ Let $\fol$ be a foliation 
with hyperbolic leaves in $M^3$
closed. Let $\Phi_1$ be a pseudo-Anosov flow almost transverse
to $\fol$ and let $\Phi$ be a corresponding almost pseudo-Anosov flow
transverse to $\fol$. Let $\Lambda^s, \Lambda^u$ be
the stable/unstable $2$-dimensional
foliations of $\Phi$ and $\wls, \wlu$
the lifts to $\mi$. Given $F$ leaf of $\fn$,
let $\wlsf, \wluf$ be the induced singular 
$1$-dimensional foliations in $F$.
Then 

\begin{itemize}

\item
For every ray $l$ in a leaf of $\wlsf$ or $\wluf$,
then $l$ limits in a single point of $\pin F$. 
%The limit point usually depends on $l$.

\item
If 
the stable/unstable foliations $\wls, \wlu$ of $\Phi$ have 
Hausdorff leaf space, then the leaves of $\wlsf, \wluf$
are uniform quasigeodesics in $F$, the bound is independent
of the leaf. In general the leaves of $\wlsf, \wluf$ are not quasigeodesic.

\item
Any non Hausdorffness (of say $\wlsf$) 
is associated to a Reeb annulus
in a leaf of $\fol$ and when projected to $M$ it either projects
to or spirals
to a Reeb annulus. 

\item
The set of ideal points of
leaves of $\wlsf$ is dense in $\pin F$ and similarly for $\wluf$.

\item
If two rays of the same leaf of $\wlsf$ limit to the same ideal
point in $\pin F$ then this leaf of 
$\wlsf$ does not contain a singularity and the region in $F$
bounded by the leaf projects in $M$ to a set in a leaf
of $\fol$ which is either contained in or asymptotic to
a Reeb annulus.
\end{itemize}

\vskip .1in
This is a completely general result:
one does not need 
negatively curved fundamental group of $M$ or
any metric properties of $\Phi$.
Theorem C is a key tool used in
the proof of the main theorem.



The article is organized as follows:
In the next section we discuss previous results
on the continuous extension property and the strategy
of the proof of the main theorem and theorem C.
In section 3 we present basic definitions and
results concerning pseudo-Anosov flows and almost
pseudo-Anosov flows. In section 4 we analyse 
projections to the orbit space.
%set $\Theta(F)$ for leaves in the universal cover.
In sections 5 and 6 we analyse the singular foliations
$\wlsf, \wluf$ and asymptotic properties of their
leaves, proving theorem C.
In section 7 we prove the continuous extension property
$-$ the main theorem.
In the final section we comment on general relationships
between foliations and Kleinian groups.

In a subsequent article we analyse other important
consequences of quasigeodesic behavior for flows
and foliations \cite{Fe10}.





%Part of this article was done while the author was
%visiting Princeton University. The author thanks this 
%institution for its hospitality and in particular
%David Gabai.


\section{Historical remarks and strategy of proofs}


Here we review what is known about the continuous extension
property.
In a seminal work, Cannon and Thurston \cite{Ca-Th} proved
this property 
when $\fol$ is a fibration over
the circle. Previously Thurston showed that a fiberting 
manifold is hyperbolic when the monodromy of the fibration
is pseudo-Anosov \cite{Th1,Th3,Th4,Bl-Ca}. Since the fundamental group
of a leaf of $\fol$ is a 
normal subgroup of the fundamental group of $M$,
then every limit set of a leaf of $\fn$ is the whole
sphere. In this way Cannon and Thurston produced many examples of 
group invariant Peano curves.

Another extremely important class of foliations is the
following:
A foliation is {\em proper} if the  leaves never limit
on themselves $-$ this is in the foliation sense and
it means that a sufficiently small transversal to a given 
leaf only
meets the leaf in a single point.
In particular leaves are not dense.
In a proper foliation there are compact leaves which
are said to have {\em depth} $0$. The depth of a leaf is
inductively defined to be $i$ (for finite $i$) if $i-1$ is the maximum
of the depths of leaves in the (foliation) limit set of
the leaf. A foliation has {\em finite depth} if it is
proper and there is a finite upper bound to the depths
of all leaves.

Gabai proved that whenever a compact $3$-manifold $M$ is 
irreducible, orientable  and
the second homology group $H_2(M, \partial M, {\bf Z})$ is not 
finite,
then there is a Reebless finite depth, foliation associated
to each non torsion homology class \cite{Ga1,Ga3}. 
The foliation is directly associated to a hierarchy of the
manifold and as such is strongly connected with the topological
structure of the manifold.
These results had several fundamental consequences for the
topology of $3$-manifolds \cite{Ga1,Ga2,Ga3}.
If $M$ is hyperbolic (or atoroidal), then
one important question is whether these finite depth
foliations have the continuous extension property.
%Conjecturally every closed hyperbolic $3$-manifold
%has a finite cover with infinite homology, to which these
%constructions apply.

%We proved  that if $\fol$ has depth one and the monodromy
%is irreducible (similar to the fibration case and it uses
%end periodic homeomorphims of non compact surfaces), then
%$\fol$ has the continuous extension property \cite{Fe1}.

Subsequently 
Gabai and Mosher showed \cite{Mo5} that any Reebless finite
depth foliation in a closed, atoroidal $3$-manifold admits
a pseudo-Anosov flow $\Phi$ which is almost transverse
to it. Roughly a flow is {\em pseudo-Anosov} if it has transverse
hyperbolic dynamics $-$ even though it may have
finitely many singularities.
It has stable and unstable two dimensional foliations which in
general are singular.
The term {\em almost transverse} means that one may need to 
blow up one singular orbit (or more) into a finite
collection of joined annuli to make the flow transverse
to the foliation. 
See detailed definitions and comments in section 3.
Since $\fol$ has a compact leaf and $M$ is
atoroidal, then Thurston \cite{Th1,Th3} proved
that $M$ is in fact hyperbolic.
%See more about pseudo-Anosov flows transverse to
%foliations below.

We proved, jointly with Mosher,
that these pseudo-Anosov flows almost transverse to finite depth 
foliations
in hyperbolic $3$-manifolds are {\em quasigeodesic} \cite{Fe-Mo}.
This means that flow lines are uniformly efficient in
measuring distance in relative homotopy classes, or 
equivalently, uniformly efficient in measuring distance
in the universal cover.
This was first proved by Mosher \cite{Mo4} for a class
of flows transverse to some examples of depth one foliations
obtained by handle constructions.
Another concept is that of quasi-isometric behavior:
a foliation (perhaps singular) is {\em quasi-isometric} if
its leaves are uniformly efficient in measuring distance
in the universal cover. There are no non singular $2$ dimensional
quasi-isometric foliations in closed $3$-manifolds
with negatively curved fundamental group
\cite{Fe1}. 
As for {\underline {singular}} foliations the situation is
quite different and there are examples.
The stable/unstable singular foliations of the quasigeodesic
flows above may be quasi-isometric \cite{Fe6} and may
not \cite{Mo5,Fe6}. 
In general quasi-isometric behavior of $\Lambda^s$ (or $\Lambda^u$)
implies that $\Phi$ is quasigeodesic.
%It depends on whether the leaf spaces
%of the foliations (in the universal cover) are Hausdorff or
%not \cite{Fe4}.


If both the stable and unstable foliations are quasi-isometric
{\underline {and}}
 the flow is actually transverse (as opposed to being
almost transverse) to the finite depth foliation then
we proved in \cite{Fe6} that $\fol$ has the continuous extension
property.
We stress that this result only applies to finite depth foliations
$-$ the proof depends, amongst other things,  on induction in the depth.
% (see 
%theorem D below for a more general result).
To use this result we needed to check
the quasi-isometric behavior of $\Lambda^s, \Lambda^u$ and
transversality between $\fol$ and $\Phi$.
This was very tricky and we could only do that for some 
{\underline {depth one}} foliations.
More to the point, it is known that  these conditions do not
always hold for general finite depth foliations.
%As remarked before, in general
%both conditions may fail. 
Corollary A proves the continuous extension property
for all finite depth foliations: there are no restrictions
on the depth of the foliation, or about transversality
of the flow with the foliation or quasi-isometric behavior
of $\Lambda^s, \Lambda^u$.


\vskip .11in
The continuous extension property was also proved for
another class of foliations:
%We mention another class of foliations.
%A {\em slithering} of $M$ is a submersion map from the universal cover
%to the circle which is group equivariant. As such it
%defines a foliation in the manifold. This foliation $\fol$
%is $\rrrr$-covered, that is, the leaf space of $\fn$ is
%%homeomorphic to the real numbers. This is equivalent
%to having Hausdorff leaf space. 
%There are many examples
%of Anosov flows in hyperbolic $3$-manifolds, so that
%the stable and unstable foliations (non singular in this case)
%are $\rrrr$-covered \cite{Fe2}.
%Up to collapsing pockets of leaves 
%the slithering property is equivalent
A foliation $\fol$ is {\em uniform} if any two leaves
in the universal cover are a bounded distance apart,  the
bound depends on the individual leaves. 
Hence $\fol$ is $\rrrr$-covered.
Thurston \cite{Th5} proved
that there is a large class of
uniform foliations.
%$\rrrr$-covered Anosov foliations, branched covering
%constructions, etc.. 
%These foliations are ${\bf R}$-{\em covered}, that is,
%the leaf space  of $\fn$ is homeomorphic to the
%real numbers. Equivalently this leaf space is Hausdorff.
If in addition
$\pi_1(M)$ is negatively
curved, then 
Thurston \cite{Th5} proved 
there is a pseudo-Anosov flow
transverse to $\fol$. From this  it is easy to 
prove  that the flow has quasi-isometric stable/unstable
foliations. In this case it also easily implies that the
foliation $\fol$ has
the continuous extension property.
The arguments are a very clever generalization of the
fibering situation.

%At this point it is important to stress 
Notice that in all the previous
results, there is a pseudo-Anosov flow $\Phi$
{\underline {transverse}} to $\fol$ and so that the
stable/unstable foliations of $\Phi$ are 
quasi-isometric singular foliations.
Both of these properties were crucial in all proofs.
The main theorem implies all the previous results
about the continuous extension property and
it has the unique feature that it applies
to an open set of foliations.


\vskip .12in
The main theorem 
can potentially be widely applicable because of
the abundance of pseudo-Anosov flows almost transverse
to foliations: Thurston proved this for fibrations \cite{Th4}.
It is also true for all
$\rrrr$-covered foliations \cite{Fe7,Cal1}
and Calegari proved it for all foliations with 
one sided branching \cite{Cal2}, all minimal foliations \cite{Cal3}
and many other foliations \cite{Cal3}.
The missing ingredient is the quasigeodesic property of
these pseudo-Anosov flows which is needed to apply
the main theorem.
In general the quasigeodesic property for a pseudo-Anosov
flow (or an arbitrary flow) is very hard to obtain.
This property is only known when the pseudo-Anosov
flow is almost transverse to
a foliation of one of the following types: 
finite depth or uniform.
One main goal in the study of a pseudo-Anosov flow $\Phi$ in
$M$ with Gromov hyperbolic fundamental group is to decide
whether $\Phi$ is quasigeodesic.
There are many examples where it is not quasigeodesic \cite{Fe2}.

\vskip .1in
Why almost transversality and not just transversality? 
In many cases ($\rrrr$-covered, 
one sided branching)  the pseudo-Anosov flow is actually
transverse to $\fol$.
But for finite depth foliations (which have
two sided branching), there are many examples where
it is impossible to make the pseudo-Anosov flow transverse
to the foliation \cite{Mo5} and one can only get almost
transversality. We will have more comments about this
in section 3.
Finite depth foliations are extremely important as they
are strongly connected to the topology of the underlying
manifold.
Also, in some sense, foliations with two sided branching
are probably more common than foliations which are
either $\rrrr$-covered or with
one sided branching. 




We also remark that 
in all previous results concerning the continuous extension
property, theorem C was 
a crucial property on which the whole analysis hinged.
%For uniform foliations this is a very easy fact.
%There is an 
%analysis of $\wlsf, \wluf$ in \cite{Fe6} for 
%finite depth foliations, with $\Lambda^s, \Lambda^u$ quasi-isometric
%and the original flow $\Phi_1$ transverse to $\fol$.
%Amongst other things the analysis is inductive on the depth
%and so of course can only work for finite depth foliations.
%In addition in that case one actually proves that
%the leaves of $\wlsf, \wluf$ are uniform quasigeodesics and
In the previous situations, the analysis of
leaves of $\wlsf, \wluf$ was either trivial or 
substantially easier.
In particular in these situations the leaves of
$\wlsf, \wluf$ were always uniform quasigeodesics, which
simplified subsequent proofs considerably. Such is
not the case in general.
In particular if $\wls, \wlu$ do not have Hausdorff leaf
space, then a priori the leaves of $\wlsf, \wluf$ do not have
to be quasigeodesic.
%this certainly is not true in many other situations.
The proof of theorem $C$ works for any pseudo-Anosov
flow almost transverse to a foliation with hyperbolic
leaves.
The proof uses the
denseness of contracting directions for foliations
as proved by Thurston \cite{Th6,Th7} when he introduced the
universal circle for foliations $-$  even though we do not
directly use the universal circle here.
The basic idea is: if any ray 
of (say) $\wlsf$ does not limit to a single
point  in $\pin F$
then it limits in a non trivial interval of $\pin F$.
Zoom into this interval and analyse the situation 
in the limit. This is actually the easiest statement to prove
in theorem C.
The facts about rays with same ideal point
and non Hausdorffness are much trickier, but they will be essential
in the analysis of the main theorem.
The results of theorem C are also used in other contexts,
for example to analyse rigidity of pseudo-Anosov flows
almost transverse to a given foliation. This will be
explored in a future article \cite{Fe10}.


The proof of the main theorem has 2 parts: given a leaf $F$ of $\fn$,
one first constructs an extension to the ideal boundary and
then show it is continuous. To define the extension, one
uses the foliations $\wlsf, \wluf$ as they hopefully
define a basis neighborhood of an ideal point $p$ of $F$.
The best situation is that the leaves
of $\wls, \wlu$ which contain these leaves
of $\wlsf, \wluf$ define basis neighborhoods of unique
points in $\si$, hence defining the image of $p$ in $\si$.
There are several difficulties here: first the leaves
of $\wlsf, \wluf$ are not quasigeodesics in $F$, so much
more care is needed. Another problem is that
the foliations $\wls, \wls$ in general do not have
Hausdorff leaf space. This keeps recurring
throughout the proof. A further difficulty is that if
intersections of leaves $L_i$ of say
$\wls$ with a leaf $F$ of $\fn$ escape
in $F$ as $i$ converges to infinity, it
does not follow that the $L_i$ themselves escape compact
sets in $\mi$.
Consequently there
are several cases to be analysed.

Another fact that is important for the analysis of the main
theorem and theorem C is the following:
Let $\Theta$ be the projection map from $\mi$ to $\oo$
(projects flow lines to points).
A leaf of $\fn$ 
intersects an orbit of $\wwp$ at most once defining an injective
projection of $F$ to $\Theta(F)$. The projection $\Theta(F)$ is equal
to $\oo$ if and only if the foliation is $\rrrr$-covered.
An important problem here is to determine the boundary 
$\Theta(F)$ as a subset of $\oo$. This turns out to be
a collection of subsets of stable/unstable leaves in $\oo$.
This result is different than what happens
for pseudo-Anosov flows transverse to foliations
and its proof is much more delicate.
This is analsyed in section 4.



\section{Preliminaries: Pseudo-Anosov flows and almost pseudo-Anosov flows}



%Pseudo-Anosov flows are a generalization of suspension flows of
%pseudo-Anosov surface homeomorphisms. These flows behave much like
%Anosov flows, but they may have finitely many singular orbits 
%which are periodic and have 
%a prescribed behavior. 


Let $\Phi$ be a flow on a closed, oriented 3-manifold $M$. We say
that $\Phi$ is a {\em pseudo-Anosov flow} if the following are
satisfied:

%\begin{itemize}

%\item 
- For each $x \in M$, the flow line $t \to \Phi(x,t)$ is $C^1$,
it is not a single point,
and the tangent vector bundle $D_t \Phi$ is $C^0$.

%\item 
- There is a finite number of periodic orbits $\{ \gamma_i \}$,
called {\em singular orbits}, such that the flow is 
``topologically" smooth off of the
singular orbits (see below).

- The flowlines are tangent to two singular transverse
foliations $\ls, \lu$ which have smooth leaves off of $\gamma_i$
and intersect exactly in the flow lines of $\Phi$.
These are like Anosov foliations off of the singular orbits.
This is the topologically smooth behavior described above.
A leaf containing a singularity is homeomorphic 
to $P \times I/f$
where $P$ is a $p$-prong in the plane and $f$ is a homeomorphism
from $P \times \{ 1 \}$ to $P \times \{ 0 \}$.
In a stable leaf, $f$ contracts towards towards
the prongs and in an unstable leaf it expands away
from the prongs.
We restrict to $p$ at least $2$, that is, we do not allow
$1$-prongs.

- In a stable leaf all orbits are forward asymptotic,
in an unstable leaf all orbits are backwards asymptotic.


Basic references for pseudo-Anosov flows are \cite{Mo3,Mo5}
and for $3$-manifolds \cite{He}.


\vskip .05in
\noindent
\underline {Notation/definition:} \ 
The singular
foliations lifted to $\mi$ are
denoted by $\wls, \wlu$.
If $x \in M$ let $W^s(x)$ denote the leaf of $\ls$ containing
$x$.  Similarly one defines $W^u(x)$
and in the
universal cover $\ws(x), \wu(x)$.
Similarly if $\alpha$ is an orbit of $\Phi$ define
$W^s(\alpha)$, etc...
Let also $\wwp$ be the lifted flow to $\mi$.




\vskip .1in
%Let $\Phi$ be a pseudo-Anosov flow in $M^3$ closed.
We review the results about the topology of
$\wls, \wlu$ that we will need.
% which will be needed in the following
%sections to prove the main theorem.
We refer to \cite{Fe4,Fe6} for detailed definitions, explanations and 
proofs.
The orbit space of $\wwp$ in
$\mi$ is homeomorphic to the plane $\rrrr^2$ \cite{Fe-Mo}
and is denoted by $\oo \cong \mi/\wwp$. 
Let $\Theta: \mi \rightarrow \oo \cong \rrrr^2$
be the projection map. 
%As the foliations $\fns, \fnu$
%are invariant under $\wwp$, they 
%induce singular, transverse $1$-dim foliations
%$\fnso, \fnuo$ in $\oo$.
%The singular points  of $\fnuo$ are the same as those
%of $\fnso$.
If $L$ is a 
leaf of $\wls$ or $\wlu$,
then $\Theta(L) \subset \oo$ is a tree which is either homeomorphic
to $\rrrr$ if $L$ is regular,
or is a union of $p$-rays all with the same starting point
if $L$ has a singular $p$-prong orbit.
%In particular every orbit in $L$ disconnects $L$.
The foliations $\wls, \wlu$ induce $1$-dimensional foliations
$\oos, \oou$ in $\oo$. Its leaves are $\Theta(L)$ as
above.
If $L$ is a leaf of $\wls$ or $\wlu$, then 
a {\em sector} is a component of $\mi - L$.
Similarly for $\oos, \oou$. 
If $B$ is any subset of $\oo$, we denote by $B \times \rrrr$
the set $\Theta^{-1}(B)$.
The same notation $B \times \rrrr$ will be used for
any subset $B$ of $\mi$: it will just be the union
of all flow lines through points of $B$.


\begin{define}
Let $L$ be a leaf of $\wls$ or $\wlu$. A slice of $L$ is 
$l \times \rrrr$ where $l$ is a properly embedded
copy of the reals in $\Theta(L)$. For instance if $L$
is regular then $L$ is its only slice. If a slice
is the boundary of a sector of $L$ then it is called
a line leaf of $L$.
If $a$ is a ray in $\Theta(L)$ then $A = a \times \rrrr$
is called a half leaf of $L$.
%The closure is denoted by $\overline A = A \cup \gamma$ and 
%its boundary is $\partial A = \gamma$.
If $\zeta$ is an open segment in $\Theta(L)$ 
it defines a {\em flow band} $L_1$ of $L$
by $L_1 = \zeta \times \rrrr$.
%\Theta^{-1}(\zeta)$.
%Let $\overline L_1$ be the closure of $L_1$ in $\mi$.
%If $\zeta$ is an open segment in $\Theta(L)$, then $\Theta^{-1}(\zeta)$
%is called a segment flow band of $L$.
Same notation for the foliations $\oos, \oou$ of $\oo$.
\end{define}


If $F \in \wls$ and $G \in \wlu$ 
then $F$ and $G$ intersect in at most one
orbit. 
Also suppose that a leaf $F \in \wls$ intersects two leaves
$G, H \in \wlu$ and so does $L \in \wls$.
Then $F, L, G, H$ form a {\em rectangle} in $\mi$
and there is  no singularity 
in the interior of the rectangle \cite{Fe6}.
There will be two generalizations of rectangles: 1) perfect fits $=$ rectangle
with one corner removed 
and 2) lozenges $=$ rectangle with two opposite corners removed.
We will also denote by rectangles, perfect fits, lozenges
and product regions the projection of these regions to
$\oo \cong \rrrr^2$. 


\begin{define}{(\cite{Fe2,Fe4,Fe6})}{}
Perfect fits -
Two leaves $F \in \wls$ and $G \in \wlu$, form
a perfect fit if $F \cap G = \emptyset$ and there
are half leaves $F_1$ of $F$ and $G_1$ of $G$ 
and also flow bands $L_1 \subset L \in \wls$ and
$H_1 \subset H \in \wlu$,
so that 
%$F_0$ is regular on the side containing $L$,
%$G_0$ is regular on the side containing $H$ and:
%
%$$ \overline L_1 \cap \overline G_1 = \partial L_1 \cap \partial G_1,
%\ \ \overline L_1 \cap \overline H_1 = \partial L_1 \cap \partial H_1,
%\ \ \overline H_1 \cap \overline F_1 = 
%\partial H_1 \cap \partial F_1,$$ 
the set 

$$\overline F_1 \cup \overline H_1 \cup 
\overline L_1 \cup \overline G_1$$

\noindent
separates $M$ and forms an a rectangle $R$ with a corner removed:
The joint structure of $\wls, \wlu$ in $R$ is that of
a rectangle with a corner orbit removed. The removed corner
corresponds to the perfect of $F$ and $G$ which 
do not intersect.
%`$${\rm with} \ \ \  \overline L_1 \cap \overline G_1 \ \not = \ \emptyset, 
%\ \ \overline L_1 \cap \overline H_1  \ \not = \ \emptyset \ \ {\rm and}
%\ \ \overline H_1 \cap \overline F_1 \ \not = \ \emptyset.$$
%\noindent
%Furthermore
%$$\forall \ S \in \wlu, \ \ \ \   
%S \cap L_1  \not = \emptyset \ \Rightarrow
%S \cap F_1  \not = \emptyset \ \ \ (1) $$
%$${\rm and} \ \ \forall \ E \in \wls, \ \ \ \ E \cap H_1  
%\not = \emptyset \ \Rightarrow
%E \cap G_1  \not = \emptyset \ \ \ (2).$$
\end{define}




We refer to fig. \ref{loz}, a for perfect fits.
%Implications $(1), (2)$ force
%equivalences (that is 
%$S \cap L_1  \not = \emptyset \ \Leftrightarrow
%S \cap F_1  \not = \emptyset$ and the same for (2)).
%The set 
%$\overline F_1 \cup \overline H_1 \cup 
%\overline L_1 \cup \overline G_1$
%separates $\mi$.
%Let $A$ be the complementary region which does
%not contain $F - F_1$ in its closure.
%An important fact is that there are 
%singularities of $\wwp$
%in $A$.
%Perfect fits produce ``ideal" rectangles, in the sense that
%even though $F$ and $G$ do not intersect, there is
%a product structure (of $\wls$ and $\wlu$) in the 
%interior of $A$.
There is a product structure in the interior of $R$: there are
two stable boundary sides and two unstable one. An unstable
leaf intersects one stable boundary side (not in the corner) if
and only if it intersects the other stable boundary side
(not in the corner).
We also say that the leaves $F, G$ are {\em asymptotic}.


%\begin{define}{\cite{Fe1,Fe4}}{}
%Given $p \in \mi$ (or $p \in \oo$), and a half leaf $H$ 
%of $\wu(p)$ defined by $\wwp_{\rrrr}(p)$, let 
%
%$$\uo(H) \ = \ \{ F \in \hs  \ | \ F \cap 
%H \not = \emptyset \} \ \subset \ \hs.$$
%
%\noindent
%Notice that $\ws(p) \not \in \uo(H)$.
%Let  also
%
%$${\cal L}^u(H)  \ = 
%\ \bigcup  \ \{ \ p \ \in \mi \ \ 
%| \ \ p \ \in \ F \ \in \ \uo(H) \ \} \ \ \subset \ \ \mi.$$
%
%\noindent
%Then ${\cal L}^u(H) \subset \mi$
%and $\ws(p) \subset \partial {\cal L}^u(H)$.
%Similarly define
%$\so(L),
%{\cal L}^s(L)$ for a stable half leaf $L$.
%\end{define}


\begin{figure}
\centeredepsfbox{gm1.eps}
\caption{a. Perfect fits in $\mi$,
b. A lozenge, c. A chain of lozenges.}
\label{loz}
\end{figure}





\begin{define}{(\cite{Fe2,Fe4,Fe6})}{}
Lozenges - A lozenge is a region of $\mi$ whose closure
is homeomorphic to a rectangle with two corners removed.
More specifically two points $p, q$ form the corners
of a lozenge if there are half leaves $A, B$ of
$\ws(p), \wu(p)$ defined by $p$
and  $C, D$ half leaves of $\ws(q), \wu(q)$ so
that $A$ and $D$ form a perfect fit and so do
$B$ and $C$. The region bounded by the lozenge
is $R$ and it does not have any singularities.
%Let $p, q \in \mi$ and half leaves
%$L_p, H_p$ of $\ws(p), \wu(p)$ defined by $\wwp_{\rrrr}(p)$,
%half leaves $L_q, H_q$ of $\ws(q), \wu(q)$
%defined by $\wwp_{\rrrr}(q)$
%%so that:
%
%
%$${\cal L}^u(L_p) \ \cap \ 
%{\cal L}^s(H_q) \ \  = \ \ 
%{\cal L}^u(L_q) \ \cap \ 
%{\cal L}^s(H_p) \ \ \subset \ \mi$$
%
%
%\noindent
%Then this intersection
%is called a lozenge ${\cal A}$
%in $\mi$, see fig. \ref{loz}, b.
%The corners of the lozenge are $\wwp_{\rrrr}(p)$ and
%$\wwp_{\rrrr}(q)$ and ${\cal A}$ 
%is a subset of $\mi$.
%The sides of ${\cal A}$ are
%$L_p, H_p, L_q, H_q$.
The sides are not contained in the lozenge,
but are in the boundary of the lozenge.
See fig. \ref{loz}, b.
\end{define}


%Sometimes we also refer to $p$ and $q$ as corners of the lozenge.

There are no singularities in the lozenges,
which implies that
$R$ is an open region in $\mi$.
There may be singular orbits
on the sides of the lozenge and the corner orbits.
%also may be singular.
%The definition of a lozenge implies that
%$L_p, H_q$ form a perfect fit and so do 
%$L_q, H_p$.
%This is an equivalent way to define a lozenge
%with corners $\wwp_{\rrrr}(p), \wwp_{\rrrr}(q)$.

%Intuitively a lozenge is a rectangle with two 
%opposite corners turned into ideal points:
%each removed corner corresponds to a pair of leaves
%(the sides of the lozenge) forming a perfect fit.

Two lozenges are {\em adjacent} if they share a corner and
there is a stable or unstable leaf
intersecting both of them, see fig. \ref{loz}, c.
Therefore they share a side.
A {\em chain of lozenges} is a collection $\{ \cc _i \}, 
i \in I$, where $I$ is an interval (finite or not) in ${\bf Z}$;
so that if $i, i+1 \in I$, then 
${\cal C}_i$ and ${\cal C}_{i+1}$ share
a corner, see fig. \ref{loz}, c.
Consecutive lozenges may be adjacent or not.
The chain is finite if $I$ is finite.


\begin{define}{}{}
Suppose $A$ is a flow band in a leaf of $\wls$.
Suppose that for each orbit $\gamma$ of $\wwp$ in $A$ there is a
half leaf $B_{\gamma}$ of $\wu(\gamma)$ defined by $\gamma$ so that: 
for any two orbits $\gamma, \beta$ in $A$ then
a stable leaf intersects $B_{\beta}$ if and only if 
it intersects $B_{\gamma}$.
%for any 
%\zeta \subset E \in \wls$ is a (possibly infinite) 
%strong stable segment 
%so that for each $p \in \zeta$ there is a half leaf $H_p$
%of $\wu(p)$ defined by $\wwp_{\rrrr}(p)$ so that
%
%$$ \forall \ p, q \in \zeta, \ \ \uo(H_p) = \uo(H_q).
%\ \ \  In \ that \ case \ let \ {\cal P} = \bigcup _{p \in \zeta} 
%H_p.$$
%
%\noindent
%Then ${\cal P} \subset \mi$ is called an unstable product
%region with base segment $\zeta$. 
%The base segment
%is not uniquely determined by ${\cal P}$.
%Similarly define negative unstable product regions 
This defines a stable product region which is the union
of the $B_{\gamma}$.
Similarly define unstable product regions.
\label{defsta}
\end{define}


%The main property of product regions is the following:
%for any $F \in \wls$, $G \in \wlu$ so that 
%$(i) \ F \cap A  \
%\not = \ \emptyset \ \ {\rm  and} \ \ 
 %(ii) \ G \cap A \ \not = \ \emptyset,
%\ \ \ {\rm then} \ \ 
%F \cap G \ \not = \ \emptyset$.
There are no singular orbits of 
$\wwp$ in $A$.

%One good way to visualize these objects in 
%$\oo$ is  as follows.
%Consider proper embeddings $\xi:U \rightarrow \oo$ of sets $U \subset 
%\rrrr^2$ into $\oo$, sending the horizontal and vertical
%foliations induced in $U$ to the stable and unstable
%foliations in $\xi(U) \subset \oo$. Then
%a proper embedding is associated to a 
%rectangle $\xi(U)$ if $U = [0,1] \times [0,1]$.
%A proper embedding is associated to a perfect fit
%if $U$ is a rectangle without a
%corner, that is, $U = [0,1] \times [0,1] - \{ (1,1) \}$. 
%A lozenge is associated to the image
%of a rectangle without two opposite
%corners $U = [0,1] \times [0,1] - \{ (1,1), (0,0) \}$
%(the lozenge is the interior of $\xi(U)$).
%A stable product region is associated to the image 
%of $U = [a,b] \times [0,\infty)$ (or $U = \rrrr \times [0,\infty)$
%when the base segment is infinite)
%and similarly for an unstable product region.
%The important fact here is that there are no
%singular orbits in any of these regions.
%

We abuse convention and call 
a leaf $L$ of $\wls$ or $\wlu$ is called {\em periodic}
if there is a non trivial covering translation
$g$ of $\mi$ with $g(L) = L$. This is equivalent
to $\pi(L)$ containing a periodic orbit of $\Phi$.
In the same way an orbit 
$\gamma$ of $\wwp$
is {\em periodic} if $\pi(\gamma)$ is a periodic orbit
of $\Phi$.


We say that two orbits $\gamma, \alpha$ of $\wwp$ 
(or the leaves $\ws(\gamma), \ws(\alpha)$)
are connected by a 
chain of lozenges $\{ {\cal C}_i \}, 1 \leq i \leq n$,
if $\gamma$ is a corner of ${\cal C}_1$ and $\alpha$ 
is a corner of ${\cal C}_n$.

If ${\cal C}$ is a lozenge with corners $\beta, \gamma$ and
$g$ is a non trivial covering translation 
leaving $\beta, \gamma$ invariant (and so also the lozenge),
then $\pi(\beta), \pi(\gamma)$ are closed orbits
of $\wwp$ which are freely homotopic to the inverse of each
other.

\begin{theorem}{\cite{Fe4,Fe6}}{}
Let $\Phi$ be a pseudo-Anosov flow in $M^3$ closed and let 
$F_0 \not = F_1 \in \wls$.
Suppose that there is a non trivial covering translation $g$
with $g(F_i) = F_i, i = 0,1$.
Let $\alpha_i, i = 0,1$ be the periodic orbits of $\wwp$
in $F_i$ so that $g(\alpha_i) = \alpha_i$.
Then $\alpha_0$ and $\alpha_1$ are connected
by a finite chain of lozenges 
$\{ {\cal C}_i \}, 1 \leq i \leq n$ and $g$
leaves invariant each lozenge 
${\cal C}_i$ as well as their corners.
\label{chain}
\end{theorem}

A chain from $\alpha_0$ to $\alpha_1$ is called
{\em minimal} if all lozenges in the chain are distinct.
Exactly as proved in \cite{Fe4} for Anosov flows,
it follows that there
is a unique minimal chain from $\alpha_0$ to $\alpha_1$
and also all other chains have to contain all the lozenges
in the minimal chain.





The main result concerning non Hausdorff behavior in the leaf spaces
of $\wls, \wlu$ is the following:

\begin{theorem}{\cite{Fe4,Fe6}}{}
Let $\Phi$ be a pseudo-Anosov flow in $M^3$. 
Suppose that $F \not = L$
are not separated in the leaf space of $\wls$.
Then the following facts happen:

\begin{itemize}

\item
$F$ is periodic and so is $L$.

\item
Let $F_0, L_0$ be the line leaves of $F, L$ which
are not separated from each other.
Let $V_0$ be the sector of $F$ bounded by
$F_0$ and containing $L$.
Let $\alpha$ be the periodic orbit in $F_0$ and
$H_0$ be the component of $(\wu(\alpha) - \alpha)$ 
contained in $V_0$.
Let $g$ be a non trivial covering translation
with $g(F_0) = F_0$, $g(H_0) = H_0$ and $g$ leaves
invariant the components of $(F_0 - \alpha)$.
Then $g(L_0) = L_0$.

\item
Let $\beta$ be the periodic orbit in $L_0$. As $g(\beta) = \beta$,
$g(\alpha) = \alpha$, then  $\pi(\alpha), \pi(\beta)$ are 
closed orbits of $\Phi$  which are  freely homotopic in $M$.
By theorem \ref{chain} \   $F_0$ and $L_0$ are connected by
a finite chain of lozenges 
$\{ A_i \}, 1 \leq i \leq n$.
%all contained in ${\cal L}^u(H_0)$k
Consecutive lozenges are adjacent.
They all intersect a common stable leaf $C$.
There is an even number of lozenges 
in the chain, see
fig. \ref{pict}.

\item
In addition 
let ${\cal B}_{F,L}$ be the set of leaves non separated
from $F$ and $L$.
Put an order in ${\cal B}_{F,L}$ as follows:
Put an orientation in the set of orbits of $C$ contained
in the union of the lozenges and their sides.
If $R_1, R_2 \in {\cal B}_{F,L}$ let $\alpha_1, \alpha_2$
be the respective periodic orbits in $R_1, R_2$. Then
$\wu(\alpha_i) \cap C \not = \emptyset$ and let 
$a_i = \wu(\alpha_i) \cap C$.
We define $R_1 < R_2$ in ${\cal B}_{F,L}$ 
if $a_1$ precedes
$a_2$ in the orientation of the set of orbits of 
$C$.
Then ${\cal B}_{F,L}$
is either order isomorphic to $\{ 1, ..., n \}$ for some
$n \in {\bf N}$; or ${\cal B}_{F,L}$ is order
isomorphic to the integers ${\bf Z}$.

\item
Also if there are $Z, S \in \wls$ so that
${\cal B}_{Z, S}$ is infinite, then there is 
an incompressible torus in $M$ transverse to 
$\Phi$. In particular $M$ cannot be atoroidal.
Also if there are $F, L$ as above, then there are
closed orbits $\alpha, \beta$ of $\Phi$ which
are freely homotopic to the inverse of each other.

\item
Up to covering translations,
there are only finitely many non Hausdorff
points in the leaf space of $\wls$.
\end{itemize}
\label{theb}
\end{theorem}


%Consider the chain ${\cal A}_i, 1 \leq i \leq n$ of
%lozenges connecting $F_0$ and $L_0$ produced
%by theorem \ref{theb}.
%Let $\alpha, \beta$ be the corner orbits of ${\cal A}_1$.
%Then $g({\cal A}_1) = {\cal A}_1$,
%$g(\alpha) = \alpha$, $g(\beta) = \beta$.
%Let $\alpha_1, \beta_1$ be the quotients of
%$\alpha, \beta$ by $g$. Then $\alpha_1, \beta_1$
%are periodic orbits of $\Phi$ - which may not
%be indivisible.
%Still their invariance by $g$ implies that $\alpha_1, \beta_1$.
%are freely homotopic.
%In fact suppose that $g$ acts as a contraction
%on the set of orbits of $\ws(\alpha)$.
%Then analysing the action of $g$ on the lozenge ${\cal A}_1$,
%we see that $g$ acts as an expansion in the set
%of orbits of $\ws(\beta)$. This implies that 
%$\alpha_1$ is freely homotopic to the inverse of
%$\beta_1$. This is fundamental for us:
%
%\vskip .03in
%\noindent
%{\underline {Conclusion}}:
%If $\wls$ is 
%not Hausdorff then there are 2 closed orbits $\gamma, \zeta$
%of $\Phi$ which are freely homotopic to the inverse
%of each other, $\gamma \cong \zeta^{-1}$.
%\vskip .03in

Notice that ${\cal B}_{F,L}$ is a discrete set in this order.
For detailed explanations and proofs, see
\cite{Fe4,Fe6}. 

\begin{figure}
\centeredepsfbox{gm2.eps}
\caption{
The correct picture between non separated
leaves of $\wls$.}
\label{pict}
\end{figure}



\begin{theorem}{(\cite{Fe6})}{}
Let $\Phi$ be a pseudo-Anosov flow. If there is
a stable or unstable product region, then $\Phi$ is 
topologically conjugate to a suspension Anosov flow.
In particular $\Phi$ is non singular.
\label{prod}
\end{theorem}


%Finally notice that product regions are very rare:
%
%\begin{theorem}{\cite{Fe4}}{}
%Let $\Phi$ be a pseudo-Anosov flow in $M^3$.
%If there is a product region
%in $\mi$, then $\Phi$ is topologically conjugate to
%a suspension Anosov flow.
%\label{produ}
%\end{theorem}


\begin{proposition}{}{}
Let $\varphi$ be a (topological) Anosov flow so that every leaf of
its stable foliation $\wls$ intersects every leaf of its stable foliations
$\wlu$. Then $\varphi$ is topologically conjugate to a suspension 
Anosov flow. In particular $M$ fibers over the circle with
fiber a torus and Anosov monodromy.
\label{susp}
\end{proposition}

\begin{proof}{}
This result is proved by Barbot \cite{Ba1} when $\varphi$
is a {\underline {smooth}} Anosov flow. That means it is $C^1$ and
it has also {\underline {strong}} stable/unstable foliations and
contraction on the level of tangent vectors along the flow.
Here we only have the weak foliations and orbits being asymptotic
in their leaves. 
%Still all the work has been done and we just
%use the pieces. 
With proper understanding all the steps carry through to the
general situation.

Lift to a finite cover where $\Lambda^s, \Lambda^u$ are transversely orientable.
A cross section in the cover projects to a cross section in
the manifold (after cut and paste following Fried \cite{Fr}) and
so we can prove the result in the cover.

First, the flow $\varphi$
is expanding: there is $\epsilon >  0$ so that no distinct 
orbits are always less than $\epsilon$ away from each other. Inaba
and Matsumoto
then proved that this flow is a topological pseudo-Anosov flow \cite{In-Ma}.
%He generalizes the ideas of Paternain ??? for homeomorphisms
%of surfaces. 
The main thing is the existence of a Markov partition
for the flow. This implies that if $F$ is a leaf of $\wls$ which
is left invariant by $g$, then there is a closed orbit of $\varphi$
in $\pi(F)$ and all orbits are asymptotic to this closed orbit.
Similarly for $\wlu$.

What this means is the following: consider the action of $\pi_1(M)$ 
in the leaf space of $\wls$ which is the reals.
Hence we have a group action in $\rrrr$. Let $g$ in $\pi_1(M)$
which fixes a point. There is $L$ in $\wls$ with $g(L) = L$.
So there is orbit $\gamma$ of $\widetilde \varphi$ with $g(\gamma) = \gamma$.
Let $U$ be the unstable leaf of $\widetilde \varphi$ with $\gamma$ contained
in $U$. Then $g(U) = U$. If $g$ is associated to the positive direction
of $\gamma$ then $g$ acts as a contraction in the set of orbits of $U$
with $\gamma$ as the only fixed point. Since every leaf
of $\wlu$ intersects every leaf of $\wls$ then the set of orbits
in $U$ is equivalent to the set of leaves of $\wls$.
This implies the important fact:

\vskip .1in
\noindent
{\underline {Conclusion}} - If $g$ is in $\pi_1(M)$ has a fixed point
in the leaf space of $\wls$ then it is of hyperbolic type and
has a single fixed point.
\vskip .1in

Using this topological characterization Barbot \cite{Ba1} showed that
$G = \pi_1(M)$ is metabelian, in fact he showed that the commutator
subgroup $[G,G]$ is abelian. In particular $\pi_1(M)$ is solvable.
This used only an action by homeomorphisms in $\rrrr$ satisfying the
conclusion above.
Barbot \cite{Ba1} also proved 
that the leaves of $\Lambda^s, \Lambda^u$ are
dense in $M$.

Plante \cite{Pl1}, showed that if $\fol$ a minimal foliation
in $\pi_1(M)$ solvable then $\fol$ is transversely affine:
there is a collection of charts $f_i:U_i \rightarrow \rrrr ^2 \times \rrrr$,
so that the transition functions are affine in the second coordinate.
Using this Plante \cite{Pl1, Pl2} constructs a homomorphism

$$C: \ \pi_1(M) \ \ \rightarrow \ \ \rrrr$$

\noindent
which measures the logarithm of how much distortion there is
along an element of $\pi_1(M)$. This is a cohomology class.
Every closed orbit $\gamma$ of $\varphi$ has a transversal fence which
is expanding - this implies that $C(\gamma)$ is positive.
Plante then refers to a criterion of Fried \cite{Fr} to conclude that
$\varphi$ has a cross section and therefore it is easily
seen that $\varphi$ is topologically conjugate
to a suspension Anosov flow. 
This finishes the proof of the proposition.
\end{proof}




\vskip .1in
\noindent
{\bf {\underline {Blown up orbits and almost pseudo-Anosov flows}}}
\vskip .05in

We now describe almost pseudo-Anosov flows.
The description is taken from \cite{Mo5}.
First we describe the blow up of a singular orbit.
Let $\gamma$ be a singular orbit of a pseudo-Anosov flow
$\varphi$ and let $C$ be a small disk transverse to $\varphi$
through a point $p$ of $\gamma$. The set  \ $\ws(\gamma)
\cup \wu(\gamma)$ \ intersects $C$ in a tree $T$, which
is made of a single common vertex and $2n$ prongs,
if the stable/unstable leaves $\ws(\gamma), \wu(\gamma)$
have $n$ prongs.
The stable and unstable prongs alternate in $C$.
Orient the prongs so that stable prongs point
towards $p$ and unstable prongs point away from
$p$. Hence $T$ is an oriented tree.
Up to topological conjugacy we may assume in 
polar coordinates centered at $p$ that 
$T \ = \ \{ (r,\theta) \ | \ \theta = k\pi/n,
k = 0,1, ..., 2n-1 \}$.

Choose a Poincar\'{e} section for $\gamma$, that is,
a smaller disk $C'$ so that every point of $C'$ flows
forward to a point in $C$. This  defines a continuous 
map $f:C' \rightarrow C$ which fixes $p$ and is a
homeomorphism into its image.
This $f$ sends the corresponding pieces of
$T$ into $T$,
contracting stable direction and expanding unstable
directions.
There is a rotation or reflection $R$ in $C$ so
that $R$ prescribes where the sectors of $C$ defined
by $T$  go to under $f$.
For instance $R$ may be a rotation by $2\pi/n$.

We first blow up $f$.
Let $D$ be a small subdisk of $C'$ containing
$p$ in its interior.
Let $T^*$ be an oriented tree in $C$ which agrees
with $T$ outside $D$ and which is invariant
under $R$ as above and such that each vertex
$v$ of $T^*$ is ``pseudo-hyperbolic", meaning
as you go around the edges incident to $v$,
the orientations of the edges of $T^*$
alternate pointing toward and away from $v$.
Each vertex of $T^*$ must have $2i$ edges for
some $i \geq 2$.
The point here is that $T^*$ is created from
$T$ by replacing $p$ with a finite subtree.
There are finitely many ways to choose
$T^*$, up to compactly supported isotopy.
The new edges of $T^*$ created
by the blow up are called the finite edges of $T^*$.


Given this data there is a $C^0$ perturbation 
$f^*$ of $f$, and a continuous map $h: C \rightarrow C$
so that 

\begin{itemize}

\item
$f^*$ leaves $T^*$ invariant.

\item
For each edge $E$ of $T^*$, the first return map of 
$f^*$ acts as a translation on
$int(E)$ moving points in the direction of the
orientation on $E$.

\item 
$h$ collapses the finite edges of $T^*$  to the
point $p$ and $h$ is otherwise $1$ to $1$.

\item
$h$ is a semiconjugacy from $f^*$ to $f$,
that is, $f \circ h = h \circ f^*$.

\item
$h$ is close to the identity map in the sup norm
and $h$ equals the identity on 
$C - D$.

\end{itemize}


\noindent
We say that  $f^*$ is obtained from
$f$ by 
{\em dynamically blowing up}
the pseudo-hyperbolic fixed point $p$.
Each choice of $T^*$ determines a unique
$f^*$ up to conjugacy by isotopy.
There are therefore finitely many ways to dynamically
blow up a pseudo-hyperbolic fixed point,
up to conjugation by isotopy.

Now define a dynamic blow up of $\gamma$ by altering
the flow $\varphi$ near $\gamma$ as follows. Let $D'$ so that
$f(D') \subset D$.
Alter $\varphi$ so that the first return map
$f: D' \rightarrow D$ is replaced
by $f^*: D' \rightarrow D$.
This has the effect of alterating the generating vector
field of $\varphi$ inside the ``mapping torus"
$T_g = \{ \varphi(x,s) \ | \ x \in D', 0 \leq s \leq t(x) \}$
and leaving the vector field unaltered outside of
$T_g$.
Let $\varphi^*$ be the new flow.


The orbit $\gamma$ gets blown up into a collection
of flow invariant annuli. In each annulus $A$, the boundary
components are closed orbits of $\varphi^*$
which are isotopic to $\gamma$
as oriented orbits.
In the interior of $A$ orbits move from one boundary to the other,
as given by the orientation of the corresponding
edge of $T^*$.
There is a blow down map $\xi:M \rightarrow M$ which is
homotopic to the identity and isotopic to the
identity in the complement of the annuli.
It sends the collection of annuli into $\gamma$.
The map $\xi$ sends orbits of $\varphi^*$ to orbits
of $\varphi$ and preserves orientation.









\begin{define}{}{}
Let $\varphi$ be a pseudo-Anosov flow in $M^3$ closed.
Then $\varphi^*$ is an almost pseudo-Anosov flow associated to
$\varphi$ if $\varphi^*$ is obtained from $\varphi$ by dynamically blowing
some singular orbits of $\varphi$.
\end{define}


%into a collection
%of flow annuli.
%Specifically if $\gamma$ is such a singular orbit of $\Phi_1$,
%then it blows up into a connected collection of annuli
%$\{ A_i, 1 \leq i \leq n \}$, each of which is flow invariant.
%The collection is embedded and the annuli have disjoint
%interiors.
%In each annulus the boundary components are closed orbits
%of $\Phi$ isotopic to $\gamma$ as oriented orbits.
%In the interior of each annulus all orbits are forward asymptotic to one
%boundary component and backwards asymptotic to the other
%one.
%There is a blow down map $\xi: M \rightarrow M$, homotopic
%to the identity and isotopic to the identity in the complement
%of the $A_i$ and sending each connected collection
%of $A_i$ into a periodic orbit of $\Phi_1$. The map
%$\xi$ sends orbits of $\Phi$ to orbits of $\Phi_1$
%preserving orientation.
%\end{define}

The reason for considering almost pseudo-Anosov flows is
as follows. All of the constructions of pseudo-Anosov flows
transverse to foliations are in fact constructions of
a pair of laminations $-$ stable and unstable $-$ which
are transverse to each other and to the foliation
\cite{Th4,Mo5,Fe7,Cal1,Cal2}.
The intersection of the laminations is oriented producing
a flow in this intersection.
One then collapses the complementary regions to the
laminations to produce transverse singular foliations and 
a pseudo-Anosov flow.
The transversality problem occurs in this last step, the
blow down of complementary regions. In certain situations,
for example for finite depth foliations, one cannot guarantee
transversality of the flow and foliation after the blow down. 
See extended explanations by Mosher in \cite{Mo5}.

The necessity of almost transversality as opposed to transversality
was first discovered by Mosher in \cite{Mo1} for positioning
surfaces with respect to certain ${\bf Z}$ invariant
flows (in non compact manifolds).
This was further explored in \cite{Mo2} where it was analysed
when it is necessary to 
blow up a pseudo-Anosov flow before
it became transverse to a given surface.


%We briefly explain
%a possible problem. Mosher's construction \cite{Mo5} of flows (almost)
%transverse to foliations is done inductively on the depth
%of the leaves (starting with the top depth leaves), associated
%to a sutured manifold hierarchy and the ensuing
%foliations construction of Gabai. At each step there is
%a foliation which is partially tangent/transverse to
%the boundary and also two laminations (stable/unstable)
%which are transverse to each other and to the foliation.
%There is a flow in the intersection of the laminations
%and a flow direction in ``periodic" leaves, since
%all orbits in say a stable leaf are forward asymptotic.
%The next step topologically involves glueing
%two subsurfaces in the boundary in the construction
%of the foliation and laminations/flow.
%
%One of the problems that can easily happen is the following.
%Suppose the glueing is done along surface $S$ and after the
%glueing there are closed orbits $\alpha, \beta$ of the flow,
%which are oriented isotopic to the same simple closed
%curve of $S$ and are in opposite sides of $S$, see
%fig. \ref{prob}.
%
%
\begin{figure}
\centeredepsfbox{gm3.eps}
\caption{
Obstruction to transversality.
Here $\alpha, \beta$ are isotopic closed orbits.
If the flow is to be pseudo-Anosov, then one has to
collapse $\alpha, \beta$ together. But then the collapsed
flow will not be transverse to the foliation.}
\label{prob}
\end{figure}


%In the resulting pseudo-Anosov flow, $\alpha, \beta$ will be
%(oriented) freely homotopic to each other. By theorem 
%\ref{chain} when lifted to
%$\mi$ they are connected
%by a finite chain of lozenges. This forces the existence
%of another closed orbit $\alpha_1$ which is freely 
%homotopic to the inverse of $\alpha$ (in the opposite corner
%of a lozenge in $\mi$).
%The problem is there is no guarantee such an orbit $\alpha_1$
%will be produced in the inductive process. In order to
%fix that, then in the collapsing step Mosher 
%collapses $\alpha$ and $\beta$ into a single orbit.
%This allows for the collapsed flow to be pseudo-Anosov.
%Unfortunately the transversality is lost locally near this
%region of $S$. There may be more collapsing forced by
%the inductive process. 
%In order to recover the transversality, in this particular
%case one blows up the collapsed orbit into an embedded
%annulus, with boundaries $\alpha, \beta$ and puts 
%a flow going from one orbit to the other, crossing
%$S$ in the correct direction. Since other collapsings
%may be forced we may have a collection of annuli which
%are joined together and collapse to a single periodic orbit.
%
%

\vskip .1in
Given a dynamic blow up $\varphi^*$ of $\varphi$ with corresponding
blow down
map $\xi$, then $\xi^{-1}$ of $\Lambda^s, \Lambda^u$ are
singular foliations, which now have some flow invariant
annuli.
These are the annuli that come from blowing up some 
singular orbits $-$ they are called the
{\em blown up annuli}.
We still denote by $\ls, \lu$ the blown up stable/unstable foliations
of an almost pseudo-Anosov flows. They are transverse
to each other except at the blown up annuli. 
The same notation is used for $\wls, \wlu$, etc..

Because $\varphi^*$ is transverse to the foliation $\fol$,
then $\wls, \wlu$ are transverse to $\fn$ and induce
singular $1$-dimensional foliations $\wlsf, \wluf$
in any leaf $F$ of $\fn$.

The objects perfect fits, lozenges, product regions, etc.. all
make sense in the setting of almost pseudo-Anosov flows: they
are just the blow ups of the same objects for the corresponding
pseudo-Anosov flows. Since the interior of these objects does
not have singularities, the blow up operation does
not affect these interiors. 
There may be singular orbits in the boundary which get
blown into a collection of annuli. All the results in
this section still hold for almost pseudo-Anosov flows,
with the blow up operation.
For example if $F, L$ in $\wls$ are not separated from
each other, then they are connected by an even number
of lozenges all intersecting a common stable leaf.
Since parts of the boundary of these may have been
blown into annuli, there is not a product structure
in the closure of the union of the lozenges, but there
is still a product structure of $\wls, \wlu$ in the interior.







\section{Projections of leaves of $\fn$ to the orbit space}

\noindent
{\bf {Notation}} $-$ In some of the proofs and arguments that
follow we will be working with an almost pseudo-Anosov flow
transverse to a foliation $\fol$. 
In those cases, for notational simplicity 
we denote the almost pseudo-Anosov flow by
$\Phi$ and a corresponding blow down pseudo-Anosov
flow by $\Phi_1$. 
Hence $\Phi$ is some blow up of $\Phi_1$.
This is different than the
notation $\varphi, \varphi^*$ from the last section.

\vskip .1in
Let then $\Phi$ be an almost pseudo-Anosov flow which is transverse
to a foliation $\fol$. 
This implies that $\fol$ is Reebless $-$ we provide
a proof of this at the end of this section.
An orbit of $\wwp$ intersects a leaf of $\fn$ at most once $-$
because the leaves of $\fn$ are properly embedded and 
$\wwp$ is transverse to $\fn$. 
Hence the projection $\Theta: F \rightarrow \Theta(F)$ is injective.
We want to determine
the set of orbits a leaf of $\fn$ intersects $-$ 
in particular we want to determine the boundary $\partial \Theta(F)$.
As it turns out, $\partial \Theta(F)$ is composed of a disjoint
union of slice leaves in $\oos, \oou$.


We assume throughout this section that $\Phi$ is {\em blow down
minimal} with respect to being transverse to $\fol$. This means
that no blow down of some flow annuli of $\Phi$ produces a flow
transverse to $\fol$.

Since $\Phi$ is transverse to $\fol$, there is $\epsilon > 0$ so that
if a leaf $F$ of $\fn$ intersects an orbit of $\wwp$ at $p$ then 
it intersects every orbit of $\wwp$ which passes $\epsilon$ near
$p$ and the intersection is also very near $p$.
To understand $\partial \Theta(F)$ one main ingredient is that 
when considering pseudo-Anosov flows, then
flow lines in the same stable leaf are forward asymptotic. So
if $F$ intersects a given orbit in a very future time then it
also intersects a lot of other orbits in the same stable
since in future time
they converge. In the limit this produces a stable boundary leaf
of $\Theta(F)$. The blow up operation disturbs this: it is not 
true that orbits in the same stable leaf of an almost pseudo-Anosov
flow are forward asymptotic: when they pass arbitrarily near a blow
up annulus the orbits are distorted and their distance
can increase  enormously. This is the key difficulty in this section.
Hence we first analyse the blow up operation more
carefully.

\vskip .1in
\noindent
{\underline {Notation}} $-$ Given $\Phi$ an almost pseudo-Anosov flow,
let 
$\Phi_1$ be a corresponding pseudo-Anosov flow associated to $\Phi$. 
The term $\ws(x)$ will denote the stable leaf of $\wwp$ or 
$\wwp_1$, where the context will make clear which one it is.
\vskip .1in


Recall that $\pi: \mi \rightarrow M$ denotes the universal covering map.


We will start with $\Phi_1$ and understand the blow up procedure.
The blown up annuli come from singular orbits.
The {\em lift annuli} are the lifts of blown up annuli
to $\mi$. Their projections to $\oo$ are called {\em
blown segments}.
If $L$ is a blown up leaf of $\wls$ or $\wlu$ the components
of $L$ minus the lift annuli are called the {\em prongs}.
A {\em quarter} associated to an orbit $\gamma$ of $\wwp_1$
is the closure of a connected component of $\mi - (\wu(\gamma)
\cup \ws(\gamma))$. Its boundary is a union of $\gamma$ and
half leaves in the stable and unstable leaves of $\gamma$.
We will be interested in a neighborhood $V$ of $\gamma$ in
this quarter which projects to $M$ near the closed orbit
$\pi(\gamma)$. 
We will understand the blow up in the projection of a
quarter. Glueing up different quarter gives the overall picture
of the blow up operation.
In the projected quarter $\pi(V)$ in $M$ there
is a cross section to the flow $\Phi_1$. The orbits across
the cross section are determined by which stable and unstable
leaf they are in. The return map on the stable direction
is a contraction and an expansion in the unstable direction.
Any contraction is topologically conjugate to say
$x \rightarrow x/2$ and an expansion is conjugate to 
$x \rightarrow 2x$. Hence the local return map is 
topologically conjugate to 



$$
\left[
\begin{array}{rr}
 1/2 & 0 \\ 0 & 2 \\ 
\end{array}
\right]$$


\noindent
a linear map. The whole discussion here is one of topological
conjugacy. The flow is conjugate to 
$(x,y,0) \ \rightarrow \ (2^{-t}x, 2^t y, t)$.
%$$V_t \ \ = \ \ 
%\left[
%\begin{array}{rrr}
% x \\ y \\ 0
%\end{array}
%\right]
%\ \ \ {\rm goes \ to} \ \ \ 
%\left[ \begin{array}{rrr}
%2^{-t}x \\ 2^t y \\ t \\
%\end{array}
%\right]$$
%
%\noindent
Think of the blow up annulus as the set of unit tangent vectors
to $\gamma$
associated to the quarter region. The flow in the
annulus is given by the action of $D V_t$ on the tangent vectors.
It has 2 closed orbits (the boundary ones corresponding to the stable
and unstable leaves). The other orbits are asymptotic to the
stable closed orbit in negative time and to the unstable
closed orbit in positive time. This makes it into a continuous flow
in this blown up part. 
%See detailed explanation in \cite{Fr} or \cite{Ha-Th} (Fried, Handel-Thurston).
For future reference recall this fact that in a blow up annulus the
boundary components are orbits of the flow and in the interior
the flow lines go from one boundary to the other without a Reeb annulus
picture (there is a cross section to the flow in the annulus).
Do this for each quarter region that is
blown up. One can then glue up the 2 sides of the appropriate annuli
because they are all of the same topological picture (using the
standard model above). This describes the blown
up operation in a quarter. There is clearly a blow down map which sends orbits
of the blown up flow $\Phi$
to orbits of $\Phi_1$ and collapses connected unions
of annuli into a single $p$-prong singular orbit.

We quantify these: let $\epsilon$ very small so that any
two orbits of $\Phi_1$ which are always less than $\epsilon$
apart in forward time, then they are in the same stable leaf.
Let ${\cal Z}$ the union of the singular orbits of 
$\Phi_1$ which are blown up.
Let $\epsilon' << \epsilon$ and let
$U$ be the $\epsilon'$ tubular neighborhood of ${\cal Z}$.
Let $U'$ (resp. $U$) be the $\epsilon'/2$ 
(resp. $\epsilon'$)
tubular neighborhood of ${\cal Z}$.
Choose the blow up map to be the identity in the complement
of $U'$, that is the blown up annuli are also contained in $U'$.
The blow down map is then 
an isometry of the Riemannian metric outside
$U'$. Choose the blow down to move points very little in $U'$.
Isotope $\fol$ so that it is transverse to the flow $\Phi$.
%We choose a very small neighborhood $U$ of the union of the blown up
%annuli and a Riemannian metric in $M$ so that the blown down operation 
%is an isometry outside of this union. Choose the blow down to be
%the identity outside this neighborhood $U$.
We are now ready to analyse $\partial \Theta(F)$.


\begin{proposition}{}{}
Let $F$ in $\fn$. Then $\Theta(F)$ is an open subset of $\oo$.
Any boundary component of $\Theta(F)$ is a slice of a leaf of 
$\oos$ or $\oou$.
If it is a slice of $\oos$, then as $\Theta(F)$ approaches $l$,
the corresponding points of $F$ escape in the positive direction.
Similarly for unstable boundary slices.
\label{bounda}
\end{proposition}

\begin{proof}{}
First notice that since $F$ is transverse to $\wwp$ then $\Theta(F)$ is
an open set. Hence $\partial \Theta(F)$ is disjoint from $\Theta(F)$.
The important thing is to notice that the metric is the same outside
the small neighborhood $U'$ of the blown up annuli. If two 
points are in the same stable leaf, then their orbits under
the blow down flow $\Phi_1$ are asymptotic in forward time.
The same is true
for $\Phi$, for big enough time if the point is outside $U$.
This is because the points of the
corresponding orbits of $\Phi_1$ will
be {\underline {both}} outside $U'$ $-$ this is the reason for
the construction of two neighborhoods $U', U$.
The following setup will be used in {\underline {all}} cases.

\vskip .1in
\noindent
{\underline {Setup}} $-$
Let $v$ in $\partial \Theta(F)$ and $v_i$ in $\Theta(F)$ with
$v_i$ converging to $v$.
Let $p_i$ in $F$ with $\Theta(p_i) = v_i$ and let $w$ in $\mi$
with $\Theta(w) = v$. Let $D$ be any small disk in $\mi$
transverse to $\wwp$ with $w$ in the interior of $D$. For $i$
big enough $v_i$ is in $\Theta(D)$ so there are 
$t_i$ real numbers with $p_i = \wwp_{t_i}(w_i)$
and $w_i$ are in $D$. As
$v$ is not in $\Theta(F)$, then $|t_i|$ grows without bound. Without loss
of generality assume up to subsequence that $t_i \rightarrow +\infty$.
Same proof if $t_i \rightarrow -\infty$.
We will prove that there is a slice leaf $L$ of $\ws(w)$ so that 
$\Theta(L) \subset 
\partial \Theta(F)$ and $F$ goes up as it ``approaches" $L$.
The stable/unstable leaves here are those of the almost pseudo-Anosov flow
and they may have blown up annuli.

One transversality fact used here is the following:
for each $\epsilon$ sufficiently small, there is $\epsilon' > 0$
so that if $x, y$ are $\epsilon$ close in $M$ then
the $\fol$ leaf through $x$ intersects the $\Phi$-orbit through
$y$ in a point $\epsilon'$ close to $x$. 
The $\epsilon'$ goes to $0$ as $\epsilon$ does. 



\vskip .1in
\noindent
{\underline {Case 1}} $-$ Suppose that $w$ is not in a blown up leaf.

First we show that we can assume no $w_i$ is in 
$\ws(w)$. Otherwise up to subsequence assume all $w_i$ are in
$\ws(w)$. The orbits through $w_i$ and $w$ start out very close
and aside from the time they stay in $\pi^{-1}(U)$ they are always very close.
%Since $F$ intersects $\wwr(w_i)$ in 
%$\wwp_{t_i}(w_i)$ then $t_i \rightarrow \infty$. 
Let $B$ be the component
of the intersection 
of $F$ with the flow band from $
\wwr(w_i)$ to $\wwr(w)$ in the stable leaf $\ws(w)$,
which contains $p_i$.
Then $B$ does not intersect $\wwr(w)$ so it has
to either escape up or down.
If it escapes down it will have to intersect a small segment
from $w_i$ to $w$ and hence so does $F$. For $i$ big enough
$w_i$ is arbitrarily near $w$, so
transversality of $\fol$ and $\Phi$ then
implies that $F$ will intersect $\wwr(w)$ near $w$, contradiction
see fig. \ref{neck}, a.

We now consider the case that $B$ escapes up. If the forward orbit
through $w$ is not always in $\pi^{-1}(U)$ then at those times outside
of $\pi^{-1}(U)$ it will be arbitrarily
close to $\wwr(w_i)$ and transversality
implies again that $F$ intersects $\wwr(w)$. If the forward orbit
of $w$ always stays in $\pi^{-1}(U)$ the same happens after the blow down
so the blow down orbit is in the stable leaf of the singular
orbit which is being blown up. 
This does not happen in case 1.
%In this particular situation
%the orbit through $w_i$ will also stay forever in $U$ once it
%enters it and will be asymptotic to $\wwr(w)$. Transversality
%rules this out. 


We can now assume that all $v_i$ are in a sector of $\oos(v)$
with $l$ the boundary of this sector and 
$L = l \times \rrrr$,
the line leaf of
$\ws(w)$ which is the boundary of this sector.

Let now $q$ in $l$. We will show that $q$ is in $\partial \Theta(F)$
so $l \subset \partial \Theta(F)$. There is a segment $[q,v]$ contained
in $l$. 
%This is disjoint from the projection of any blown up annuli.
Choose $x$ in $L$ with $\Theta(x) = q$. Let $\alpha$ be a segment
in $\ws(w)$ transverse to the flow lines and going from $x$ to $w$. 
%Since
%$\ws(w)$ has no blown up annulus we may choose a neighborhood
%$N$ of $\alpha$ which is disjoint from the lift annuli.
%This is because the blown annuli in $M$ form a compact set.
Notice that $x, w$ are in $\mi$ and $q, v$ are in the orbit
space $\oo$.
Let $x_i$ converging to $x$ and
$x_i$ in $\ws(w_i)$. We can do that since all $w_i$
are in the same sector of $\ws(w)$.
 Choose segments $\alpha_i$ from $x_i$ to $w_i$ in 
$\ws(w_i)$ and transverse to the flowlines of $\wwp$ in $\ws(w_i)$.
%For $i$ big enough $\alpha_i$ is contained in $N$ and
%so $\alpha_i$ does not intersect lift annuli. This implies
%that $\wwr(\alpha_i)$ is disjoint from lift annuli.

\vskip .1in
\noindent
{\underline {Claim}} $-$ For every orbit $\gamma$ 
of $\wwp$ intersecting $\alpha_i$
in $y$ then $\gamma$ intersects $F$ in $\wwp_s(y)$ where
$s$ converges to $\infty$ as $i \rightarrow \infty$.

%The orbits through $y$ and $w_i$ will get arbitrarily close in forward time
%whenever they are outside $\pi^{-1}(U)$. 
%Because the $v_i$ are in a sector
%of $\Theta^s(v)$ they are not in $\ws(v)$. 
%Therefore the point $y$ is not in $\ws(w)$ and so its future orbit
%will not stay in $U$ forever. 
%For $i$ big enough $w_i$ is not in $\pi^{-1}(U)$???
%FOr $i$ big enough the segments $\alpha_i$ are
%disjoint from lift annuli. Otherwise $\alpha$ would
%intersect a lift annulus because             
%the blown annuli in $M$ form a compact set. Hence the
%flow band $\wwr(\alpha_i)$ does not intersect lift annuli.

Suppose there is $a_0 > 0$ so that for some $i_0$ then

$$\wwp_{[a_0,t_i]}(w_i) \ \subset \ \pi^{-1}(U)
\ \ \ {\rm for \ all} \ \ i \geq i_0$$

\noindent
Then $\wwp_{[a_0,\infty)}(w)$ is contained in the closure
of $\pi^{-1}(U)$.
As seen before this implies that $w$ is in a blown up
stable leaf, which is not the hypothesis of case 1.
Therefore up to subsequence, 
there are arbitrary
big times $s_i$ between $0$ and $t_i$ so that $\wwp_{s_i}(w_i)$ is
not in $\pi^{-1}(U)$. 
Hence $\wwr(x_i)$ is very close to $\wwp_{s_i}(w_i)$
and since $F$ cannot escape up or down then $F$ intersects 
$\wwr(x_i)$. Hence the segment
$[\Theta(x_i), v_i]$ of $\oos(v)$
is contained in $\Theta(F)$ and
so $[x,v]$ is contained in the closure of $\Theta(F)$. Also
the time $s$ so that $\wwp_s(y)$ hits $F$ goes to $\infty$, hence
$[x,v]$ cannot intersect $\Theta(F)$ $-$ else there would
be bounded times where it intersects $F$, by transversality.
We conclude that 
$[x,v] \subset \partial \Theta(F)$, hence $l \subset \partial \Theta(F)$
as desired.
If there is a sequence $z_i$ in $F$
escaping down with $\Theta(z_i)$ converging to 
a point in $l$, then by connectedness
there is
one intersecting a compact middle region $-$ this would force
an intersection of $F$ with $l \times \rrrr$ which is impossible.

This finishes the proof of case 1.
In this case we proved there is a {\underline {line leaf}} $l$
of $\Theta(L)$ with $l \subset \partial \Theta(F)$ and $F$ escapes
up
as $\Theta(F)$ approaches $l$.



\begin{figure}
\centeredepsfbox{gm4.eps}
\caption{
a. A strangling neck is being forced,
b. A slice in a leaf of $\oos$ or $\oou$.
$x_i = \Theta(z_i)$.}
\label{neck}
\end{figure}



\vskip .1in
\noindent
{\underline {Case 2}} $-$ $w$ is in a blown up leaf, but
$F$ does not intersect a lift annulus in $\ws(w)$.

%Here we split into several situation which together give the whole picture.
%Some of the previous analysis carries over to this case as well.
%Suppose first that $p \times \rrrr$ is not in  
%a lift annulus. 
%The analysis in case 1 can be used here.
%We
%use the same notation as in case 1, with $v_i, w_i, p_i = 
%\wwp_{t_i}(w_i), t_i \rightarrow \infty$. 

Refer to the setup above.
As before we first show we can assume $w_i$ are not in $\ws(w)$.
Otherwise, up to subsequence assume all $w_i$ are in
$\ws(w)$.
Since $F$ does not intersect lift annuli in $\ws(w)$, then
$w_i$ are all in prongs of $\ws(w)$. Up to subsequence
we can assume they
are all 
in the same prong $C$ of $\ws(w)$ which has boundary an orbit
$\gamma$ of $\wwp$.
It follows that $w$ is in $\gamma$.
All the orbits in $C$ are forward asymptotic to $\gamma$,
even in the blown up situation.
The strangling necks analysis of case 1 shows that $F$
will be forced to intersect $\wwr(w)$.
This cannot occur.


Hence assume all $v_i$ are in a sector of $\oos(v)$ bounded
by a line leaf $l$. Let $L$ be $l \times \rrrr$.
%The half leaf $L$ does not contain any lift annulus.
%If $w_i$ are in $\ws(p)$ then since $t_i \rightarrow \infty$,
%then $F$ intersects $\wwr(p)$, contradiction. So assume $v_i$
%are in a sector of $\ws(p)$. 
Let $q$ be a point in $l$ and choose $x, \alpha,
x_i$ and $\alpha_i$ as in the proof of case 1.
Choose a small disc $D$ which is transverse to $\wwp$ and
has $\alpha$ in its interior. For $i$ big enough  then
$D$ intersects lift annuli  only in $\ws(x)$.
This is because the union of the blown annuli forms a compact set
in $M$, so  either $\alpha$ intersects a lift annulus,
in which case there is no other lift annulus nearby
or $D$ is entirely disjoint from lift annuli.
%If the forward orbits of $w_i$ are always in $\pi^{-1}(U)$ then
%it is in a prong of a blown up stable leaf and the proof
%of case 1 applies otherwise 
From now on the arguments of case 1 apply perfectly.
This shows that 
$\Theta(L)$ is contained in $\overline {\Theta(F)}$,
it is disjoint from $\Theta(F)$ and so it is $\partial \Theta(F)$
and $F$ escapes up as it approaches $L$.
This finishes the proof of case 2.

%This gives a partial picture and not a slice leaf in
%the boundary $\Theta(F)$. Now we need to understand what happens
%when $F$ intersects lift annuli.



Now we need to understand what happens when $F$ intersects
a lift annulus in general. We separate that in a special case.
We need the following facts before addressing this case.
A lift annulus $W$ through $b$ is contained in $\ws(b)$ and
$\wu(b)$ so there is not stable/unstable flow directions
in $W$. However there are still such directions
in $\partial W$, because one attracts nearby orbits
of $\wwp$ in $W$ and the other one repels nearby orbits
in $W$. In
this generalized sense the first one is stable and the
second one is unstable. In this sense
if $a$ is in an endpoint of a blown segment,
then all local components of $\oos(a) - a, \oou(a) - a$ near $a$
are either generalized stable or unstable. With
this understanding there is an even number of such
components and they alternate between generalized stable and unstable.
Some local components of $\oos(a) - a$ are also local
components of $\oou(a) -a$ if they are blown segments.
One key thing to remember is that generalized 
stable and unstable alternate.

\vskip .1in
\noindent
{\underline {Case 3}} $-$
Suppose that $F$ intersects some lift annulus $A$ contained
in $\ws(w)$.

Then 
$F$ does not intersect both boundary orbits of $A$.
Otherwise collapse $\pi(A)$ to a single orbit, still
keeping $\Phi$ transverse to $\fol$. 
This contradicts that $\Phi$ is blow down minimal with
respect to $\fol$.
Hence either $F \cap A$ is contained in
the interior of $A$
or it intersects only one boundary leaf.


Assume without loss of generality that $F$ escapes 
{\underline {up}} in one
direction.
% $-$ which then has to be getting closer to
%he stable direction.
This defines an orbit $\gamma$ of $\wwp$ with
$a = \Theta(\gamma)$ in $\partial \Theta(F)$.
The orbit $\gamma$ has to be in the boundary of the lift
annulus $A$. This is because an interior orbit is asymptotic
to both boundary orbits and hence would intersect $F$.
We now look at the picture in $\oo$. Consider the stable leaf
$\oos(a)$. Notice that $\Theta(F)$ intersects $\Theta(A)$. 
%In 
%$\oos(a)$ there
%are two adjacent stable sides to $\Theta(A)$. 
From the point
of view of $\gamma$, orbits in $A$ move away from $\gamma$ in
future time, that is $A$ is an unstable direction
from $\gamma$. This means that $\Theta(A)$ is
generalized unstable as seen from $a$.
It follows that 
there are two generalized stable sides of $\oos(a)$
one on each side of $\Theta(A)$
which are the closest to $\Theta(A)$.
Choose one side, start at $a$ and follow along $\oos(a)$ if needed
through blown segments and eventually into a prong in 
$\oos(a)$ so as to
produce a piece of a line leaf of $\oos(a)$ in that direction.
This path is regular on the side associated
to $\Theta(A)$ and defines a half leaf
$l_1$ of $\oos(a)$. Similarly define
$l_2$ in the other direction, see fig. \ref{neck}, b.
Let $l$ be the union of $l_1$ and $l_2$. Then $l$ is a slice leaf
of $\oos(a)$ but is not a line leaf since $\Theta(A)$ is in $\oos(a)$
and is not in $l$.
%Notice that $l$ is not a line leaf $-$ in order for that to
%happen $l_1$ would have to be followed by $\Theta(A)$ for instance
%and not by $l_2$.


\vskip .1in
\noindent
{\underline {Claim}} $-$ $l$ is contained in $\partial \Theta(F)$
and $F$ escapes positively as $\Theta(F)$ approaches $l$.

Let $b$ in $l_1$ with $b$ not in blown segment, that is,
$b$ in a prong.
Choose  $b_i$ in $\oou(b)$, with 
$b_i \rightarrow b$ and in that component of $\oo - l$.
Let $D$ be an embedded disc in $\mi$ which is transverse to $\wwp$ and
projects to $\oo$ to a neighborhood of the arc $\xi$ in $l_1$ 
from $a$ to $b$.
Let $y_i$ in $D$ with 
$\Theta(y_i) = b_i$, $y_i \rightarrow y$ with $\Theta(y) = b$.
Assume that $y$ is not in $\pi^{-1}(U)$.
Choose $b$ so that it is not in the unstable leaf of one singular orbit,
hence $\wu(y)$ does not contain lift annuli.
In addition choose $y_i$ so that $\ws(y_i)$ does not contain lift annuli either.

Choose points $u_j$ in $F \cap A$ so that 
$\Theta(u_j) = a_j$ converges to $a$.
For each $j$ the set $\Theta(F)$ contains a small neighborhood
$V_j$ of $\Theta(u_j)$ with $V_j$ converging to $a$ when
$j$ converges to infinity. 
%Notice that  
%$F$ intersects $A$ transversely. 
%From $u_j$ slide along $F$ until one gets to the complement of $\pi^{-1}(U)$.
%This defines an arc $\beta_j$ in $F$. 
Notice that $a$ is not in $V_j$ as $a$ is not in $\Theta(F)$.
The leaves $\oos(b_i)$ are getting closer 
and closer to $l_1$ and $\Theta(A)$.
For $j$ fixed there is $i$ big enough so that
$\oos(b_i)$ intersects $V_j$. Let

$$z_i \ \in \ F \cap \ws(y_i) \ \ {\rm with} \ \ 
\Theta(z_i) \in V_j$$

\noindent
here $i$ depends on $j$.
%leaves $\ws(y_i)$ are getting closer and closer to 
%the blown up strip $A$.
%Then for $i, j$ big enough the arc
%$\beta_j$
%will
%intersect $\ws(y_i)$ producing $z_i$ in $F$ with 
%$z_i$ in $\ws(y_i)$.  
Let $z_i = \wwp_{t_i}(r_i)$ with $r_i$ in $D$. 
%Then $t_i$ converges to 
%infinity.
By choosing $j$ and $i$ converging
to infinity we get that 
%If we fix $j$ big enough and allow $i$ to grow then we 
%can assure that 
$\Theta(z_i)$ converges to $a$ and we can ensure that the
arc of  $D \cap \ws(y_i)$  between $r_i$ and $y_i$ is converging
to the arc $\eta$ of $\ws(\gamma) \cap D$ with $\Theta(\eta) = \xi$.
We can also choose $V_j$ small enough so that $t_i$ converges to
$+\infty$.
%in $l_1$.
%Now from $y_i$ go to the point
%$w_i$ in $\wwr(z_i)$ in this segment. 

The orbits $\wwr(y_i), \wwr(r_i)$ are very close 
in the forward direction as long
as they are outside $\pi^{-1}(U)$. Since $\ws(y_i)$ does not
contain lift annuli then for times $s$
converging to infinity $\wwp_s(y_i)$ is not in $\pi^{-1}(U)$.
%Also $\wwp_s(y_i)$ is not in $\pi^{-1}(U)$ for $s$ small
%as $y$ is not in this set.
Consider the flow band $C$ in $\ws(y_i)$
between $\wwr(r_i)$ and $\wwr(y_i)$. 
The leaf $F$ intersects $\wwr(r_i)$ in $\wwp_{t_i}(r_i)$ with 
$t_i$ converging to infinity. 
Then an analysis exactly as in case 1 considering strangling necks
and the arcs $B$ in that proof, shows that $F \cap \ws(y_i)$ 
cannot escape up 
before intersecting $\wwr(y_i)$. 

Suppose that $F$ escapes down before intersecting $\wwr(y_i)$.
We show that this is impossible.
Since $F \cap \ws(y_i)$ has points $z_i$ in the forward
direction from $D$ and points in the backwards direction
from $D$ it follows that
$F \cap \ws(y_i)$ must intersect  $D$ in at least a
point $q_i$. Up to subsequence we may assume that $q_i$
converges to $q$ in $\ws(y)$.
This will be an iterative process. 
Let $u_1 = q$.
It is crucial
to notice that in the flow band of $\ws(y)$ between $\wwr(y)$ and
$\gamma$ the flow lines tend to go closer to $\gamma$,
that is, either they project to closed orbits freely
homotopic to $\pi(\gamma)$ or they are asymptotic
to one of these orbits moving closer to $\gamma$.
We now consider the component
of $F \cap \ws(y)$ containing $u_1$ and follow
it towards $\gamma$. 
This component does not intersect $\gamma$ and by
the above it can only escape down in $\ws(y)$.
As it escapes down it produces points $c_i$ in $\ws(r_i)$ 
and as before produces points $c'_i$ in $D$, which
up to subsequence converge to $c$ in $D \cap F$.
By construction $c$ is not $u_1$ and its
orbit is closer to $\gamma$.
Let $u_2 = c$. 
%Notice that $u_2$ is different than $u_1$
%by construction.
We can iterate this process. 
Notice the $u_i$ cannot accumulate in $D$, or else
all the corresponding points of $F$ are in a compact
set of $\mi$.
On the other hand the process does not terminate.
This produces a contradiction.



The contradiction shows that in fact  
the arc
$\Theta(C)$ is in $\Theta(F)$ which implies that 
$\xi = \Theta(\eta)$ is
contained in $\overline {\Theta(F)}$. As the time to hit $F$
from $D$ grows with $i$, this shows that $\Theta(F)$ does
not intersect $\xi$ and hence $\xi$ is contained in $\partial \Theta(F)$.
As $b$ is arbitrary this shows that $l \subset \partial \Theta(F)$ and
$F$ escapes up as $\Theta(F)$ approaches $l$.
This finishes the analysis of case 3.



%\vskip .1in
%For $i$ big enough the region in $\ws(y_i)$ between $\wwr(y_i)$ and
%$\wwr(z_i)$ does not intersect a blown up annulus, because
%in $M$ the collection of blown up annuli is compact and they
%are flow invariant.
%Therefore $\ws(y_i)$ is not always in $\pi^{-1}(U)$ and whenever it
%is not, it will be very close to $\wwr(z_i)$. Now an analysis as in
%case 1, cannot escape up or down applies and it follows
%that the segment in $\oos(x_i)$ between $x_i$ and a point
%very close to $v$ is in $\Theta(F)$ with times very positive.
%These orbits are
%asymptotic in future time. As $z_i$ is outside of
%$U$, then so is the corresponding point of $\wwr(y_i)$ and
%so the entire segment from $z_i$ to $x_i$ is in $\Theta(F)$. As
%$x_i$ converges to $x$ and $z_i$ converges to $v$, it follows
%that $l_1 \subset \overline \Theta(F)$. Since the times
%go to infinity then $l_1 \subset \partial \Theta(F)$. 
%Similarly for $l_2$. So $l = l_1 \cup l_2$ is contained
%in $\partial \Theta(F)$. 




%\vskip .1in
%We continue the analysis of case 2.1.
%The segment $\Theta(A)$ in $\oo$ has boundary two points
%$v, c$. We now look at the behavior of $\Theta(F)$ beyond $c$.
%If $F$ does not intersect $c \times \rrrr$ we produce
%more boundary $\partial \Theta(F)$, in this case $F$
%going down as it approaches
%$c \times \rrrr$. Let $b$ in $c \times \rrrr$.
%If $F$ does intersect $\wwr(b)$ then it can intersect
%other lift annuli $B_1,...B_n$ with a boundary component
%in $\wwr(b)$. They cannot intersect the other boundary
%component of $B_i$ producing stable or unstable boundary
%as in the beginning of case 3.1.

%This analysis shows also that the point $p$ in $\partial \Theta(F)$
%cannot be an orbit in the interior of a lift annulus.
%This is because of the transversality of $\fn$ and the lift
%annuli, so if $p$ corresponds to an interior orbit
%then $F$ in fact intersects the interior of the lift annulus.
%As seen in the beginning analysis of case 2.1 then $\Theta(F)$
%can only have boundary points in the boundary points of the
%projection of the lift annulus.


%If on the other hand the $w_i$ escape down, then
%$l = l_1 \cup l_2$ is also a slice in the boundary and $l$ escapes
%down. Now if $F$ intersects an annulus, then in at least one
%direction 
%one obtains a boundary component of $\Theta(F)$ as we desired.

%If we go in the other direction with respect to blown 
%$F$ can intersect at most
%one more blown annulus, forming a chain of 2. Whenever
%it intersects another blown annulus we produce more 
%$\partial \Theta(F)$ either up or down.

%GIven the analysis above if $F$ intersects a blown up 
%annulus with $v$ in it, then $v$ is in the boundary
%of the annulus, there is a slice $l$ in $\oos(v)$
%with $l \subset \partial \Theta(F)$ and $F$ goes
%up as it approaches $l$.



\vskip .1in
\noindent
{\underline {Case 4}} $-$ $w$ is in a blown up stable leaf and
$F$ intersects some lift annulus $A$ in $\ws(w)$.

The difference from case 3 is that in case 3 we obtained a slice
boundary $l$ of $\Theta(F)$ $-$ but in our situation 
we do not yet know
if it contains $\Theta(w)$ and  whether it is a stable or
unstable. Here we prove it is a stable slice and it contains
$\Theta(w)$.

Recall the setup: $v = \Theta(w)$ is in $\partial \Theta(F)$
and there are $v_i$ in $\Theta(F)$ with $v_i$ converging  to 
$v$ and with $p_i$ in $(v_i \times \rrrr) \cap F$.
Also $p_i = \wwp_{t_i}(w_i)$ with $w_i$ converging to $w$
in $\mi$ and $t_i$ converging to infinity.
%In case 3 we assume that $F$ intersects lift annulus $A$ in
%$\ws(w)$.
Let $\xi$ be the blown segment $\Theta(A)$.

The analysis of case 3 shows that $\Theta(F)$ contains 
the interior of $\Theta(A)$.
Suppose first that 
$v$ is in $\xi$. Then $v$ is in the boundary of 
$\xi$ and by case 3 again $F$ escapes up
or down when $\Theta(F)$ approaches a slice
which contains $v$.
If it escapes up, then the slice is a stable slice
and we obtain the desired result in this case.
We now show that $F$ does not escape down. 
Let $l$ be the unstable slice in $\partial \Theta(F)$
associated to this. Then $l$ cuts in half a small
disk neighborhood of $v$ in $\oo$.
The set $\Theta(F)$ intersects only one component
of the complement, the one which intersects $\xi$.
As $F$ escapes down when $\Theta(F)$ approaches $l$, then
for all points in $\Theta(F)$ near enough $v$ the corresponding
point in $F$ is flow backwards from $D$.
This contradicts the
fact that $t_i$ is converging to infinity.
Therefore $F$ cannot escape down as it approaches $l$.

We can now assume that $v$ is not in $\xi$.
By changing $\xi$ if necessary assume that $\xi$ is the
blown segment in $\oos(v)$  intersected by $\Theta(F)$ which is
closest to $v$.
Let $z$ be the endpoint of $\xi$ separating the rest of $\xi$
from $v$ in $\oos(v)$. 


We first show that $z$ is not in $\Theta(F)$. Suppose
that is not the case and let $b$ the intersection point
of 
$z \times \rrrr$ and $F$.
Since $\xi$ is the last blown segment of
$\oos(v)$ between $\xi$ and $v$ intersected by
$\Theta(F)$ and $\Theta(F)$ contains an open neighborhood
of $z$, it follows that $v$ is in a prong $B$ of
$\oos(v)$ with endpoint $z$. 
Let $\tau$ be the component of $F \cap \ws(b)$ containing
$b$. Since $F$ does not intersect
$v \times \rrrr$ then it escapes. As the region between
$\wwr(b)$  and $z \times \rrrr$ (should it be
$v \times \rrrr$ instead of $z \times \rrrr$? previous
report???)
is a prong,
then $F$ cannot escape up. As seen in the arguments
for case 3, $F$ cannot escape down either.
This shows that $z$ cannot be in $\Theta(F)$.

%Let $\beta$ be
%the segment of $\oos(v)$ from $z$ to $v$.
%Let $D$ be a disk in $\mi$ transverse to $\wwp$
%so that $\Theta(D)$ is a small neighborhood of $\beta$.
%Since $\beta$ is in the closure of a prong of $\oos(v)$ then 
%$\oos(v_i)$ limits on $\beta$ as $i$ converges
%to infinity $-$ whether $v_i$ is in $\oos(v)$ or not.
%So for big enough $i$ there are 
%
%$$r_i \ \in \ D \cap \ws(w_i), \ \ \ \Theta(r_i) \rightarrow z
%\ \ {\rm as} \ \ i \rightarrow \infty$$
%
%\noindent
%Let $\alpha_i$ be the segments in $\oos(v_i)$ between 
%$v_i$ and $\Theta(r_i)$. As seen in case 1 we can choose
%$r_i$ so that for $i$ big enough $\alpha_i$ does not
%intersect blown segments. Then the forward part of
%the orbits $\wwr(r_i)$ and $\wwr(w_i)$ are very close
%while outside $\pi^{-1}(U)$. As seen in 
%the proof of case 1 there are $s_i$ converging
%to infinity with $\wwp_{s_i}(r_i)$ in $F$. But this
%contradicts the existence of bounded $a_i$ with
%$\wwp_{a_i}(r_i)$ in $F$ near $b$. 

It follows that $F$ escapes either up or down as $\Theta(F)$
approaches $z$. 
Suppose first that it escapes up.
Then we are in the situation
of case 3 and we produce a stable slice $l$ in
$\partial \Theta(F)$ with $F$ going up as $\Theta(F)$ approaches
$l$. 
If $v$ is not in $l$ then $l$ separates $v$ from
$\Theta(F)$. This contradicts $v_i$ in $\Theta(F)$
with $v_i$ converging to $v$. Hence $v$ is in $l$
with $F$ escaping up as $\Theta(F)$ approaches $l$.
This is exactly what we want finishing the analysis
in this case.

The last situation is $F$ escaping down in $A$ as $\Theta(F)$
approaches $z$. By case 3 there is a slice leaf $l$ in
$\oou(z)$ with $l$ contained in $\partial \Theta(F)$ and
$F$ escaping down as $\Theta(F)$ approaches $l$.
We want to show that this case cannot happen.
Notice that the blown segments 
of $\oos(z)$ are exactly the
same as the blown segments of $\oou(z)$. The sets $\oos(z), \oou(z)$
differ exactly in the prongs and as they go around the collection
of blown segments. 
The collection of all prongs in $\oos(z), \oou(z)$
also alternates between stable and unstable as it goes around
the union of the blown segments.

Suppose first that $v$ is in $l$.
This contradicts $F$ escaping down and $t_i \rightarrow \infty$.
Finally suppose that $v$ is not in $l$.
We claim that in this case $l$ separates $v$ from $\Theta(F)$.
Let $\alpha$ be the path in $\oos(v)$ from $z$ to $v$.
If $\alpha$ only intersects $l$ in $z$, then the separation
property follows because
$l_1$ and $l_2$ contain the local components
of $\oos(z) \cup \oou(z) - z$ which are closest to $\Theta(A)$.
%see fig. \ref{bar}, a. 
This was part of the construction
of $l$ in case 3.
Here the $\xi$ is generalized stable at $z$
and $l_1, l_2$ are generalized unstable at $p$. The path from $z$ to 
$v$ in $\oos(v)$ cannot start in $\xi$ or $l_1$ or $l_2$, hence
$l$ separates $\Theta(F)$ from $v$.

%\blankfig{bar}{1.1}{a. The case $\alpha \cap l$ is only $z$,
%b. The case when $\alpha \cap l$ is a nontrivial subsegment.}

If on the other hand $\alpha \cap l = \delta$ is not
a single point, then it is a union of blown segments.
Let $u$ be the other endpoint of $\delta$. By regularity
of $l_1$ and $l_2$ on the $\Theta(F)$ side it follows that each
blown up segment in $\delta$ has flow direction away from
$z$. Hence $\delta$ is generalized stable at $u$.
Therefore the closest component of
$\oos(u) \cup \oou(u) - u$ on the
$\Theta(F)$ side is generalized unstable and that is contained
in $l$.
%, see fig. \ref{bar}, b.
In this case it also follows that $l$ separates $v$ from
$\Theta(F)$. As seen before this is a contradiction.

This finishes the proof of proposition \ref{bounda}
\end{proof}

%Suppose first there is no blown segment in $\oos(v)$ between
%$\xi$ and $v$.
%Then if $v$ is in a blown up segment if follows that
%$v$ is in $l$ and $F$ ha
%Then if $v$ is in a blown up segment if follows that
%$v$ is in $l$. But notice that $F$ escapes down as $\Theta(F)$
%approaches $l$ and so $\Theta(F)$ does not intersect any other
%%component of $\oo - l$.
%Since $F$ escapes down and $t_i$ converge to infinity this
%is a contradiction.
%The other option is that $v$ is in a prong $B$ of $\oos(v)$
%with endpoint $z$, see fig. \ref{bar}, a.
%Locally near $\xi \cup B$ there is a sector $W$ of $\oo$ that $\Theta(F)$
%intersects. There is $B'$ which is either a prong of $\oou(v)$
%or a blown segment of $\oou(v)$ which cuts the sector, separating
%$\xi$ from $v$, see fig. \ref{bar}, a.
%Then $\xi \cup B'$ is contained in $l$ and hence in $\partial \Theta(F)$
%and this shows that $v$ cannot be in the closure
%of $\Theta(F)$. This is a contradiction.
%
%
%
%The other option is that there is a blown segment
%$\xi_1$ in $\oos(v)$ separating $\xi$ from $v$, see 
%fig. \ref{bar}, b. Let $W$ be the corresponding sector
%as the situation above. Notice there can be no unstable
%prong of $\oou(v)$ entering $w$ or the separation argument
%above applies. Then as seen in case 3.1 it follows that
%$\xi_1$ is also contained in $l$, that is $\xi_1$ is
%in $\partial \Theta(F)$ and $F$ escapes down.
%Let $z_1$ be the other endpoint of $\xi_1$. 
%Then we can restart the analysis with $z_1$ instead
%of $z$. Induction shows this cannot happen.
%
%This finishes the proof of proposition????
%







%
%
 %with $p$ in $\partial \Theta(F)$
%and $F$ not intersecting any blown up annulus in $\ws(v)$.
%Assume now that $F$ does not intersect a blown up
%annulus with $v$ in it. Then locally $F$ is in a sector
%of $\ws(p)$. If $v$ is in the interior of a blown up
%annulus with direction $a_2 \rightarrow a_1$. We follow
%backwards from $a_2$ along the blown up annuli $A_1, A_2$
%until one gets to a half leaf $L_1$. Here $F$ does not
%intersect $A_1$. If it intersects $A_2$ then
%it does not intersect the contracting end so it
%escapes down. By the analysis done previously (Case 2.1)
%$F$ escapes down along a slice leaf of $\oou(a_2)$.
%This contradicts $F$ limiting up to $v$.
%In this case we have a continuous set of leaves, not
%just a sequence of leaves.
%
%Finally we check whether $F$ intersects $L_1$. 
%Since it does not intersect the boundary it escapes and
%it needs to escape down.
%As in the proof of case 2.1 it produces a line leaf in $\oos(p)$
%towards which $F$ escapes down, contradiction.
%Either separates the $v_i$ or escapes up.
%
%We obtain that $F$ does not intersect that half leaf on
%that side and neither in the other side as well.
%Suppose the unstable leaf coming into $a_1$ is a blown
%up annulus and $F$ intersects it. Then it goes up
%as it nears $a_1$ and the slice leaf $l_1 \cup l_2$ is in
%the boundary of $F$ as required. Otherwise now 
%$\oos(p) \cap F = \emptyset$ and since there is no
%blown up annulus separating any 2 points in this
%line leaf of $\ws(p)$ we obtain that this line leaf
%of $\oos(p)$ is contained in $\partial \Theta(F)$
%and $F$ is going up as it approaches this line leaf.
%
%If now there are $p_i$ in $F$ with $\Theta(p_i) \rightarrow v$
%and $p_i = \wwp_{t_i}(w_i)$ with $w_i \rightarrow w$ in 
%$\mi$ and $t_i \rightarrow -\infty$, then the whole
%analysis produces an unstable slice in $\oos(v)$ 
%which is entirely in the boundary of $\Theta(F)$. These
%two possibilities are incompatible. Either the slice
%in $\oou(F)$ cuts the slice of $\oos(F)$ into 2 pieces
%in different components of $\mi - R$ impossible since they
%have to limit in 2 sides. The other option is that both
%of them are on the same side. But then the stable slice cuts
%$F$ away from $\partial \Theta(F)$ as seen in the unstable
%slice. Hence we can only have $t_i \rightarrow \infty$.
%Also $F$ really escapes up as it approaches this slice.
%
%This finishes the proof of proposition boundary of $\Theta(F)$.
%%\end{proof}





%\section{Projections}
%\label{proj}
%
%Given $F$ a leaf of $\gn$ we want to determine the set of orbits of 
%$wwp$ it intersects of $\oo(F)$.
%
%\begin{lemma}{}{}
%Given $F$ in $\gn$, it intersects an orbit of $\wwp$ at most once.
%This means that $\Theta: R \rightarrow \oo$ is injective
%for any $F$ in $\gn$.
%\end{lemma}
%
%\begin{proof}{}
%Otherwise one can perturb a segment of $\wwp$ jointly with a segment in
%$F$ to produce a closed transversal to $\gn$.
%\end{proof}
%
%\begin{proposition}{}{}
%Let $F$ in $\gn$ and $a$ a point in the boundary of $\Theta(F)$
%(that is $\partial \Theta(F)$. Then either there is a subleaf $l$ of
%$\wu(a)$ or $\ws(a)$ so that $\Theta(l)$ is contained in
%$\partial \Theta(F)$ and $l$ does not intersect $F$.
%\end{proposition}
%
%\begin{proof}{}
%Let $a_i$ be points in $\oo$, with $a_i$ in $\Theta(F)$ and $a_i$ converges
%to $a$.  Consider a small disk $D$ in $\mi$ transverse to $\wwp$
%with $a$ in $D$. Let $b_i$ in $D$ with $\theta(b_i) = a_i$. 
%Since $a_i$ are in $\Theta(F)$, there are points $p_i$ in $F$ with
%$p_i$ in $\wwp_{t_i} (b_i)$. 
%Up to a subsequence suppose that $t_i$ converges to $+\infty$.
%
%\vskip .1in
%\noindent
%{\underline {Case 1}} - Suppose that $v$ is not in the lift of
%a blown up annulus.
%
%Hence $\ws(v), wu(v)$ neatly divides the neighborhood of 
%$v$ in sectors. Up to subsequences assume all $v_i$ are
%in the closure of the same sector.  If $v_i$ is in $\ws(v)$, then
%since $t_i \rightarrow \infty$ it follows that $v_i$ is very 
%close to $\wwp_{\rrrr}(v)$ so as $F$
%is transverse to $\wwp$ then $F$ intersects $\wwp_{\rrrr}(v)$
%contradiction. 
%
%If now $r$ is in $l$ and
%$\wu(r) \cap ws(l)$ intersects for $i$ big enough in a point the orbit
%of $b_i$. For $i$ big enough???
%$\Phi_{t_i}(b_i)$ is very close to $wwp_{t_i}(v_i)$ so
%$\wwp_{t_i}(b_i)$ is in $F$. Therefore $r$ is in the closure
%$\Theta(F)$ in $\oo$. 
%
%If on the other hand $r$ is in $\Theta(F)$ there is a small disk $D$ in
%$F$ transverse to $\wwp$ so that $r$ is in $D$. Then for $i$ big enough
%$b_i$ is in $D$ contradiction to the previous lemma.
%
%\vskip .1in
%\noindent
%{\underline {Case 2}} - 
%The point $r$ is in a blown up annulus
%and $r$ is an interior point.
%
%Notice holonomy is as picture in figure 1.
%Notice in the interior of the picture is a product of the
%leaf shoots throuh, hence this case cannot happen.
%
%\vskip 1.5in
%\centerline{figure 1 will go here}
%
%\end{proof}

This has an important consequence that will be used extensively
in this article.

\begin{proposition}{}{}
Let $F$ in $\fn$ and $L$ in $\wls$ or $\wlu$.  Then the intersection
$F \cap L$ is connected.
\label{conn}
\end{proposition}

\begin{proof}{}
By transversality of $\fol$ and
$\Phi$, the intersection $C = \Theta(F) \cap \Theta(L)$
is open in $\Theta(L)$.
Suppose there are 2 disjoint components $A, B$ of $C$.
Then there is $v$ in $\partial A$ with $v$ separating
$A$ from $B$. There are $v_i$ in $A$ with $v_i$ converging
to $v$. By the previous proposition $F$ escapes up or down
in $A \times \rrrr$ as $\Theta(F)$ approaches $v$.
Assume wlog that $F$ escapes up. Then there is a slice
leaf $l$ of $\oos(v)$ with $l \subset \partial \Theta(F)$
and $F$ escapes up as $\Theta(F)$ approaches $l$.
Since $l$ and $\Theta(F)$ are disjoint then $B$ is
disjoint from $l$. In addition $v$ separates
$B$ from $A$ in $\Theta(L)$. It follows from the construction
of the slice $l$ as being the closest to $A$, that
$l$ separates $A$ from $B$. Hence $\Theta(F)$ cannot
intersect $B$, contrary to assumption.
This finishes the proof.
\end{proof}




As promised, we now prove that $\fol$ being almost 
transverse to a pseudo-Anosov flow implies that $\fol$
is Reebless.

\begin{proposition}{}{}
Let $\fol$ be a foliation almost transverse to a pseudo-Anosov
flow  $\Phi_1$ and transverse to a corresponding almost pseudo-Anosov
flow $\Phi$. Then $\fol$ is Reebless.
\end{proposition}

\begin{proof}{}
Suppose that $\fol$ is not Reebless and consider a Reeb component
which is a solid torus $V$ bounded by a torus $T$.
Assume that the flow $\Phi$ is incoming along $T$.

Recall that  there are some singular orbits of $\Phi_1$
which blow up into a collection of flow annuli of
$\Phi$. Suppose that $V$ intersects one of these annuli
$A$. Then since $\Phi$ is incoming along $T$, the torus
$T$ cannot intersect the closed orbits in $\partial A$.
Hence it intersects the interior of $A$, say in
a point $p$ and the forward orbit of $p$ will
limit in a closed orbit which is contained in
the interior of $V$.

If on the other hand $V$ does not intersect these blown
annuli then the blow down operation does not affect
the flow in $V$. That means we can assume that $\Phi_1$ is
equal to $\Phi$ in $V$. Since orbits of $\Phi_1$
are trapped inside $V$
once they enter $V$, the shadow lemma for pseudo-Anosov
flows \cite{Han,Man,Mo1}, shows that 
there is also a periodic orbit of $\Phi_1$ (and hence
also of $\Phi$) in $V - T$.
Notice that the shadow lemma is for pseudo-Anosov flows
and not for almost pseudo-Anosov flows and that is why
we split the analysis into 2 cases.

In any case there is a closed orbit $\gamma$ of $\Phi$ contained
in the interior of $V$.
Consider the generalized stable/unstable local
leaves at $\gamma$. Since $\Phi$ is incoming along $T$,
the generalized unstable leaves have to be contained
in $V$. We eventually obtain that a whole half leaf
of $W^u(\gamma)$ is contained in $V$. 
A lift $\widetilde V$ to $\mi$ is homeomorphic to $D^2 \times
\rrrr$, because closed orbits of $\Phi$ are not null homotopic.
The procedure above produces a half leaf of $\wu(\widetilde \gamma)$
contained in $\widetilde V$. This contradicts the
fact that $\wu(\widetilde \gamma)$ is properly 
embedded \cite{Ga-Oe}.
This shows that $\fol$ is Reebless.
\end{proof}




\section{Asymptotic properties in leaves of the foliation}
\label{asym}

Let $\Phi$ be an almost pseudo-Anosov flow transverse
to a foliation $\fol$ with hyperbolic leaves. Let
$\Lambda^s, \Lambda^u$ be the singular foliations
of $\Phi$. Given leaf $F$ of $\fn$ let $\wlsf, \wluf$
be the induced one dimensional singular foliations in $F$.
In this section we study asymptotic properties of 
rays in $\wlsf$.
First we mention a result of Thurston \cite{Th5,Th7} concerning
contracting directions, which for convenience we
state for 3-manifolds:

\begin{theorem}{}{(Thurston)} 
Let $\fol$ be a codimension one foliation with hyperbolic
leaves in $M^3$ closed. Then for every $x$ in any leaf
$F$ of $\fn$ 
and every $\epsilon > 0$ there is a dense set of
geodesic rays 
of $F$ starting at $x$ such that:
for any such ray $r$
there is a transversal $\beta$ to $\fn$ starting at $x$ so
that any leaf $L$ intersecting $\beta$ and any
$y$ in $r$, then the distance between $y$ and $L$ is
less than $\epsilon$.
If there is not a holonomy invariant transverse measure
whose support contains $\pi(F)$ then one can show
that the directions are actually contracting, that is:
if $y$ escapes in $r$ then the distance between $y$ and
$L$ converges to $0$.
Finally if $\pi(F)$ is not closed one can choose the 
$\beta$ above to have $x$ in the interior.
\end{theorem}

There is a carefully written published version of this
result in \cite{Ca-Du}.
The directions above where distance to nearby $L$ goes to
$0$ are called contracting directions. The other ones
where distance is bounded by $\epsilon$
are called $\epsilon$ non expanding directions.

The goal of this section is to show that given a leaf
$L$ of $\fn$ and a ray $l$ of ${\widetilde \Lambda}^s_L$, 
then $l$ converges to a single point in $\pin L$.
We first analyse 
the non $\rrrr$-covered case.
%if $l$ does not 
%converge to a single point in $\pin L$, this forces
%$\fol$ to be $\rrrr$-covered. 
The proof is very involved and is done by way 
of contradiction.
Later we deal
with the $\rrrr$-covered situation.
This result is a natural extension of a result by
Levitt  \cite{Le} who proved that if
$\gal$ is a foliation with prong singularities
in a closed hyperbolic surface, then in the universal
cover, an arbitrary ray converges to a single point in
the circle at infinity.
The situation for non compact leaves of foliations
is much more delicate.





\begin{proposition}{}{}
Let $\Phi$ be an almost pseudo-Anosov flow transverse to
a foliation $\fol$ in $M^3$ closed and $\fol$ with hyperbolic
leaves. 
%Let $\varphi$ be the blow up flow transverse to
%$\la$. 
Suppose that $\fol$ is not $\rrrr$-covered.
Given a leaf $L$ of $\fn$ and an arbitrary ray $l$ 
in a leaf of ${\widetilde \Lambda}^s_L$ or
${\widetilde \Lambda}^u_L$ 
then $l$ limits to a single
point in $\pin L$. The limit depends on the ray $l$.
\label{as1}
\end{proposition}

\begin{proof}{}
We do the proof for 
${\widetilde \Lambda}^s_L$.
Let $\epsilon$ positive so that if $p$ in $\mi$ is
less than $\epsilon$ from a leaf $F$ of $\fn$, then
the flow line through $p$ intersects $F$ less than
$2 \epsilon$ away from $p$.
Let $l$ be a ray in ${\widetilde \Lambda}^s_L$.
Because $\fol$ and $\Phi$ are transverse,
$L$ is properly embedded in $\mi$ and leaves of $\wls$ are
properly embedded, it follows that $l$ is a properly embedded
ray in $L$. Therefore it can only limit in 
$\pin L$. 

Suppose by way of contradiction that $l$ 
limits on 2 distinct points $a_0, b_0$ in $\pin L$.
Fix $p$ a basepoint in $L$. 
Since
$l$ limits in $a_0, b_0$, there are compact arcs $l_i$ of
$l$ with endpoints which converge to $a_0, b_0$ respectively
in
$L \cup \pin L$ and so that the distance from $l_i$ to $p$ in
$L$ converges to infinity. Also we can assume that the
$l_i$ converges to a segment $v$ in $\pin L$, where $v$ connects
$a_0, b_0$. This is in the Hausdorff topology of closed sets
in $L \cup \pin L$,  which is a closed disk.

The key idea is to bring this situation to a compact part of $\mi$.
Choose a sequence $p_i$ at bounded distance
from points in $l_{k_i}$ so that that $p_i$ converges
to a point $a$ in the interior of $v$. 
The bound depends on the sequence.
Up to subquence
assume that there are convering translations $g_i$ in 
$\pi_1(M)$ and a point $p_0$ in $\mi$ 
so that $g_i(p_i)$ converges to 
$p_0$ in $\mi$.

We claim that the set of possible limits $p_0$ 
obtained as above projects
to a sublamination of $\fol$.
Clearly if $g_i(p_i)$ converges to $p_0$ and $q$ is in the same 
leaf $L_0$ of $\fn$ as $p_0$, then the distance 
from $p_0$ to $q$ is finite 
and there are $q_i$ in
$L$ with $d_L(q_i, p_i)$ bounded and
$g_i(q_i)$ converging to $q$. Also $q_i$ converges to $a$ in $\pin L$.
In addition if a sequence of such limits $c_j$ converges
to $c_0$ then a diagonal process shows that $c_0$ is also
obtained as a single limit.
This proves the claim.
Choose a minimal sublamination $\ccl$.



A leaf $F$ of $\fn$ is isometric to the hyperbolic plane. A 
{\em wedge} $W$ in $F$ with corner $b$ and ideal set 
an interval $B \subset \pin F$
is the union of the rays in $F$ from $b$ with ideal point in $B$.
The angle of the wedge is the angle that the boundary rays of
$W$ make at $b$.
For any such sequence $p_i$ as above, then
the visual angle at $p_i$ subintended by the arc $v$ in $\pin L$ grows
to $2 \pi$. Therefore the angle of wedge with corner
$p_i$ and ideal set 
$\pin L - v$ converges to $0$.
This is called the {\em bad wedge}.



Assume up to subsequence that $g_i(p_i)$ is converging
to $p_0$ in a leaf $L_0$ of $\fn$ and that the 
directions of the
bad wedges with corners $g_i(p_i)$ in $g_i(L)$ are converging 
to the direction $r_0$  of $L_0$. Let $c$ be the ideal
point of $r_0$ in $\pin L_0$.

Suppose first that $\pi(L_0)$ is not compact $-$ we shall
see briefly that this is in fact always the case.
Thurston's theorem shows that the set of two sided contracting
directions (or $\epsilon$ non
expanding directions) in $L_0$ is dense
in $\pin L_0$. 
We will use these to transport a lot of the structure
of ${\widetilde \Lambda}^s_{L_0}$ to nearby leaves.
%Choose $F_1, F_2$ on both sides of
%$L_0$ so that there is a big disk $D$ in
%$L_0$ which is less than $\epsilon$ from both
%$F_0$ and $F_1$.
%Very near $r_0$.
Choose $s_0, s_1$ to be rays in $L_0$ defining
contracting directions
(or $\epsilon$ non expanding directions)
very near $r_0$ so that together they form a small wedge
$W$ in $L_0$ with corner $p_0$.
There is an interval of leaves near $L_0$ so that
any such leaf $V$ is less than $\epsilon$ away from 
$s_0, s_1$. Then a flow line  of $\wwp$ through any point
in $s_0$ or $s_1$ intersects $V$ less than $2 \epsilon$
away. So $s_0$ flows to a curve in $V$, where we can
assume it has geodesic curvature very close to $0$,
if $\epsilon$ is sufficiently small. It is therefore 
a quasigeodesic with a well defined ideal point.
The same happens for $s_1$ and the flow images $u_0, u_1$
of $s_0, s_1$ in $V$ define a generalized wedge $W'$
in $V$.
%For $i$ big enough $g_i(p_i)$ is very close to $p_0$ and
%$g_i(L)$ is in this interval so $s_0, s_1$ flow into
%$g_i(L)$ with images 
The ideal points $e_0, e_1$ of $u_0, u_1$  are
close and bound an interval $I$ which is 
almost all of $\pin g_i(L)$.


By construction $g_i(l)$ is a
ray which limits in an interval of $\pin g_i(L)$ which
contains $I$ in its interior if
$i$ is big enough. 
There are then  subarcs $\tau_j$ of $g_i(l)$ with endpoints
$a_j, b_j$ in $u_0, u_1$ respectively so that $a_j$ converges
to $e_0$ and $b_j$ converges to $e_1$ and $\tau_j$ converges
to $I$, see fig. \ref{inde}.
Here $i$ is fixed and $j$ varies. 
Since $a_j, b_j$ are in $u_0, u_1$ then they flow 
(by $\wwp$) to points
in $L_0$. The images in $L_0$ are in the same leaf of  $\wls$.
By proposition \ref{conn} these images 
are in the same leaf of
$\widetilde \Lambda^s_{L_0}$.
Hence the whole segment $\tau_j$   flows into
$L_0$.  


\begin{figure}
\centeredepsfbox{gm5.eps}
\caption{
Transporting the
structure between leaves $g_i(L)$ and $L_0$.}
\label{inde}
\end{figure}



The point $p_0$ flows into $p'$ in $g_i(L)$ under the flow.
The arc $\tau_j$ together with 
subarcs or $u_0, u_1$ from $a_j, b_j$ to $p'$ 
bound a disc $D_j$ in $g_i(L)$. The arguments above show
that the boundary of $D_j$ flows into $L_0$ producing
a curve in $L_0$ bounding 
a disc $B_j$. The segments of $\wwp$ connecting points
in $\partial D_j$ to points in
$\partial B_j$ produce an annulus $C_j$. Then $D_j \cup C_j
\cup B_j$ is an embedded sphere in $\mi$ and hence
bounds an embedded ball. 
Since orbits of $\wwp$ are properly embedded in $\mi$,
it follows that all orbits of $\wwp$ intersecting
$D_j$ will also intersect $B_j$. Hence there is product
flow in this ball. Since this is true for all $j$ then
the union of the $D_j$ flows into $L_0$.
The union of the $D_j$ is the closure of $g_i(L) - W'$.
The image is contained in the closure of $L_0 - W$
in $L_0$ $-$ call the closure $J$. 

We claim
that the image is in fact $J$.
All the $\tau_j$ are in the same leaf of $\wls$ and hence
all their flow images  in $L_0$ also are. 
Since rays of $\widetilde \Lambda^s_{L_0}$ are
properly embedded in $L_0$ then 
when $j$ converges to infinity the images of $\tau_j$
in $L_0$ escape
compact sets. This shows the claim.
Therefore the flow produces a homeomorphism between
the closure of $L_0 - W$ and the closure of 
$g_i(L) - W'$. Clearly the same is true for any leaf
in the interval
associated to the contracting (non expanding) directions
$s_0, s_1$. 
In particular we have the following conclusions:

\vskip .1in
\noindent
{\bf {Conclusion}} $-$ In any limit leaf $L_0$ with 
a limit direction $r_0$ of bad wedges the following
happens: Let $c$ be the ideal point of $r_0$ and
$A$ a closed interval of $\pin L_0 - \{ c \}$.
Then there is a leaf $l$ of $\widetilde {\Lambda}^s_{L_0}$
with compact subsegments $l_i$ so that the endpoints
of $l_i$ converge to the endpoints $a, b$ of
$A$ and $l_i$ converges to $A$.
In particular $l_i$ escapes compact sets.
There are also subsegments $v_i$ with both endpoints
converging to $a$ and so that $v_i$ converges to 
sets in $\pin L_0$ which contain $A$.
Finally for sufficiently near leaves 
there is a wedge in $L_0$ which forms
a product flow region with these nearby leaves.
\vskip .1in

To get the second assertion above just follow $l$ beyond the 
endpoint of $l_i$ near $b$ until it returns near $a$
again.
As a preliminary step to obtain proposition \ref{as1} we prove
the following:







\begin{lemma}{}{}
%Either $\fol$ is $\rrrr$-covered or
For any limit $g_i(p_i)$ converging to $p_0$,
the distinguished direction of the bad wedge associated
to $g_i(p_i)$ converges to a single direction at $p_0$.
In the second case
this direction varies continuously with the leaves in $
\widetilde {\ccl}$.
\label{single}
\end{lemma}


\begin{proof}{}
Suppose there are subsequences $q_i, p_i$ converging to 
points in (interior) $v$ with $g_i(p_i), h_i(q_i) \rightarrow p_0
\in L_0 \in \fn$,
but the directions of the wedges converge to $r_0, r_1$
distinct geodesic rays in $L_0$.
We will first show that there is an interval of leaves
of $\fn$  so that the flow $\wwp$ is a product flow 
in this region.

Using the limit direction $r_0$ we produce a wedge
$W$ in $L_0$ so that the closure of $L_0 - W$
is part of a product flow region with nearby leaves
of $\fn$.
Using the other limit direction $r_1$
we produce a flow product region associated to
another wedge region $W_*$ disjoint from $W - p_0$.
Together they produce a global product structure
of the flow in a neighborhood of $L_0$.








%there are contracting directions (or non expanding
%directions $s_0, s_1$ so that for all sufficiently near to them
%$F$ is $\epsilon$ near $L$ and so flow lines intersecting
%$L_0$ in $s_0, s_1$ will intersect $F_1$ as well. Furthermore since
%$g_i(l_k)$ (here with $k >> i$) is an arc very close to $\pin L_0 - 
%\overline{s_0 s_1}$ then this also goes through from $L_0$ to $L_1$.

%The set of points obtained as above forms a sublamination of $\Gamma$.
%Clearly id $g_i(p_i)$ converge to $p_0$ and $q$ is in the same 
%leaf $L_0$ as $p$, then the distance is finite and there are $q_i$ in
%$L$ with $d_L(q_i, p_i)$ bounded and
%$g_i(q_i)$ converging to $q$. Also $q_i$ converges to $v$ in $L$.
%So the set of limits is a sublamination. We extract a minimal sublamination
%$\lal$ out of it.

%The region bounded by $g_i(l_k)$ and compact arcs of $s_1, s_0$
%from $p_0$ to $s_1 \cap g_i(l_k)$, and $s_0 \cap g_i(l_k)$ has boundary
%which goes through from $L_0$ to $L_1$. In $g_i(L)$ it also bounds
%a disk $D$. Then $D_0 \cup W \cup D_1$ is a sphere which 
%bounds a ball $B$ in $\mi$. Since orbits of $\widetilde \varphi$ are
%properly embedded, it 
%follows that every orbit intersecting $D_0$ must intersect
%$D_1$ and vice versa. 

%The only thing missing from this 
%picture is the small wedge defined by $s_0$ and
%$s_1$. But if we have another sequence converging to another direction
%it will eat up the wedge. 

%Notice this depends only on the 
%picture in $L_0$ and not in the limiting leaves.
%%
%\vskip .1in
%\noindent
%{\bf {Conclusion}} - 

This shows that there is a neighborhood $N$ of $L_0$ in the
leaf space of $\fn$ so that
the flow is a product flow in $N$.
In particular there is no non Hausdorffness of $\fn$ in
this neighborhood.
This is a very strong property as we shall see below. It 
implies a global product structure of the flow.

Notice that the structure of $\widetilde \Lambda^s_{g_i(L)}$ in
$g_i(L) - W'$ flows over to $L_0$. In particular there are 
many rays of $\widetilde \Lambda^s_{L_0}$ which do not
have a single limit in $\pin L_0$. This implies that
$\pi(L_0)$ is not compact. This is because Levitt \cite{Le}
proved that given any singular foliation with prong singularities
in a closed hyperbolic surface $R$, then 
the rays of the lift to $\widetilde R$ all have unique
limit points in the ideal boundary.
This shows that the minimal lamination $\ccl$ is not a compact leaf
and hence it has no compact leaves.


\vskip .1in
Consider the neighborhood $N$ as above.
Consider the translates $g(N)$ where $g$ runs through all elements
of the fundamental group. 
Let $P$ be the component of the union containing $N$. 
It is easy to see that the
set $P$ is precisely invariant: if $g$ is in $\pi_1(M)$
and $g(P)$ intersects $P$ then $g(P)$ is equal to $P$.
In addition $\fol$ restricted to $P$ has leaf space
homeomorphic to $\rrrr$ because of the product flow
property.
We are assuming that $N$ is open.

Suppose first that $P$ is not all  of $\mi$, hence
$\partial P$ is a non empty collection of leaves of $\fn$.
Let $C$ be the projection of $P$ to $M$.
Then $C$ is open, saturated by leaves of $\fol$.
Notice that $g(P)$ does not intersect $\partial P$ for
any $g$ in $\pi_1(M)$ for otherwise $g(P)$ intersects
$P$ and so $g(P) = P$. It follows that $\pi(\partial P)$
is disjoint from $C$ hence $C$ is a proper open, foliated
subset of $M$.

Dippolito \cite{Di} developed a theory of such open, saturated
subsets. Let $\overline C$ be the metric completion of $C$.
There is an induced foliation in $\overline C$, which
we will also denote by $\fol$. Then

$$\overline C \ \ = \ \ V \ \cup \ \bigcup_1^n V_i$$

\noindent
where $V$ is compact and may be all of $\overline C$.
Each nonempty $V_i$ is an $I$-bundle over a non compact
surface with boundary, so that $\fol$ is a foliation
transverse to the $I$-fibers. Each component of
the intersection $\partial V_i \cap V$ is an annulus
(or Moebius band) with induced foliation transverse
to the $I$ fibers.
In our situation with $\Phi$ transverse to the flow,
if $V$ is not $\overline C$, we can choose $V$
big enough so that the flow is transverse to $\fol$ 
in each $V_i$ and induces an $I$-fibration there.


Let $\widetilde R$ be a component of $\partial P$ and
$R$ the projection to $M$, so $R$ is component
of $\partial C$.
%Suppose first that $R$ is closed. We show this
%is impossible, basically using holonomy.
Parametrize the leaves of $\fn$ in $P$ as
$F_t, 0 < t < 1$ with $t$ increasing with
flow direction. A leaf in the boundary of $P$ which
is the limit of leaves in $P$ which are limiting from
the positive side above has to be the
limit of $F_t$ as $t$ goes to $0$: 
Suppose that $S$ is in the boundary of $P$ and there
are $x_i$ in $F_{t_i}$ with $t_i$ converging
to $t_0 > 0$ 
and $x_i$ converging to $x$ in $S$.
Then $S$ and $F_{t_0}$ are not separated from
each other.
For $i$ big enough the flow line
through $x_i$ will intersect $S$ 
and therefore this flow line will not intersect
$F_{t_0}$. This contradicts the fact
that  $F_{t_o}$ and $F_{t_i}$ 
have a flow product structure.

Suppose then that $\widetilde R$ is a limit of $F_t$
where $t$ converges to $0$.
Suppose first that $R$ is compact. 
%Let $x$ be
%a point in $\widetilde R$ and $\gamma$ be the
%flow line of $\wwp$ through $p$. Then $\gamma$ intersects
%$F_t$ for $t$ close to $0$ hence intersects all $F_t$.
Suppose there are $t_i$ converging to $0$ so that
$F_{t_i}$ are in $\widetilde {\ccl}$. Then since
$\ccl$ is a closed subset of $M$ it follows that
$\widetilde R$ is in $\widetilde {\ccl}$ and so
$R$ is in $\ccl$. But $R$ is closed, contradicting
the fact that $\ccl$ has no closed leaves.
There is then $a > 0$ which is the smallest $a$ 
so that $F_a$ is in $\widetilde {\ccl}$ $-$ notice
that $\widetilde {\ccl}$ has leaves in $P$.
For any $g$ in $\pi_1(R)$ then $g(N) \cap N$ is
not empty hence $g(N) = N$. It follows that
$g(F_a) = F_b$ for
some $b$. If $b$ is not $a$ then by taking
$g^{-1}$ if necessary we may assume that $b < a$.
But as $F_b$ is in $\widetilde {\ccl}$, this contradicts
the definition of $a$. Hence $g(F_a) = F_a$ for
any $g$ in $\pi_1(R)$. 
This implies that $\pi(F_a)$ is a closed surface,
again contradiction.

We conclude that $R$ is not compact, hence it eventually
enters some $V_i$ (the point here is that $V$ is
not $\overline C$). 
The flow restricted to any component of $\partial V_i
\cap \overline C$ goes from one component to the
other in the annulus. This implies that all $\pi(F_t)$ 
intersect this annulus. 
There is then a leaf $B$ of $\ccl$ which enters
$V_i$. Going deeper and deeper in this non compact $I$-bundle
will produce a limit point which is not in $C$.
This shows the very important fact that $\ccl$ is
not contained in $C$ and therefore

$${\cal E} \ = \ \ccl  \cap   (M - C) \ \not  =  \ \emptyset$$


\noindent
In addition ${\cal E}$ is not equal to $\ccl$ 
since 
$\ccl$ has leaves in $C$
and
$(M - C)$ is closed.
Hence ${\cal E}$ is a non trivial, proper sublamination
of  $\ccl$. This contradicts the fact that $\ccl$ is
a minimal lamination.

This shows that the assumption $P \not = \mi$ is
impossible. Hence $P = \mi$, which implies the
flow $\wwp$ produces a global product picture
of $\fn$ and in particular $\fol$ is $\rrrr$-covered,
contrary to assumption.

This shows 
%that if $\fol$ is not $\rrrr$-covered, then
the limits of the bad wedges are unique directions
in the limit leaves. It also shows that they vary
continuously from leaf to leaf, for otherwise one
obtains bad wedges in very near leaves which have
definitely separated directions. The same proof above
then applies.
This finishes the proof of lemma \ref{single}.
\end{proof}


%If $g_1(N)$ intersects $g_2(N)$, then
%their union is a region where there is a product flow. Obviously the
%foliation restricted to these unions has Hausdorff leaf space, homeomorphic
%to an interval in the reals.
%Notice that the structure in any of these product regions, has the same structure
%as the original interval.
%This means that the two ideal points are distinguished and in a product 
%region this carries through. 
%Let $P$ be a maximal product region. Consider the top boundary.
%If the top has more than one leaf in the limit, let $F_1, F_2$ be 
%leaves in the limit. Take a lot of contracting directions in $F_i$
%evenly spaced in $F_i$. The theory of transverse directions
%shows that there is a unique direction associated to $b_1$ in 
%$F_1$ and similarly for $b_2$, see figure ????
%
%\vskip 1.5in
%\centerline{Figure 3, the non Hausdorff situation}
%
%If they are distinct then the previous argument shows that all of 
%$F_1$ flows into $Z$. If they are the same, we get that
%more and more of $F_1$ flows in to $Z$ as $Z$ gets closer
%and closer to $F_1$. Since the $N$ is a product region, then
%all of $F_1$ flows into $Z$ for all $Z$ and vice versa. 
%
%This shows that there can be no other leaf of $\lal$ non
%separated from $F_1$ and the flow is a product in
%$N \cup F_1$. 
%
%	If the distinguished points in $F_1$ are distinct, we can
%iterate this process. There is a small neighborhood of $F_1$ in
%the leaf space where we can extend the product picture with 2 distinct ideal
%points. 
%
%\vskip .2in
%MINIMALITY - aside from previous page?????
%
%We take $\lal^*$ to be a minimal sublamination.
%
%\vskip .1in
%\noindent
%{\underline {Claim}} - $\lal^*$ is not a closed leaf.
%
%If $\lal^*$ is a closed surface $R$, then $\wls_R$ is a singular
%foliation in a closed surface. Levitt \cite{Le} has studied
%this in detail and showed that every leaf converges to
%a single point in $\pin R$. This cannot occur in our situation.
%\vskip .1in
%
%
%Going in the positive direction, this process may either take up
%all the leaves in the positive direction. This proves that $\fol$
%above $L_0$ is $\rrrr$-covered. This implies that $\fol$ in
%an $\rrrr$-covered foliation and finishes the proof in this
%case.
%Otherwise there is a maximal $N_1$ with boundaries $F_0, F_1$
%where the ideal points are identified.
%The interval defined by $F_0, F_1$ and $N_1$ is precisely invariant:
%either $g(I) = I$ or $g(I) \cap I = \emptyset$. For otherwise there is
%$g$ in $\pi_1(M)$ with $g(F_0)$ in $N_1$ 	or $g(F_1)$ in $N_1$.
%
%%	Either the top leaf has a unique bad point or there are at	least 2. 
%%If there are at least 2, we can continue the process, otherwise
%it is precisely invariant. Then $N_1$ projects to 
%an open foliated subset $O$ of $M$ with 
%$\pi(F_0), \pi(F_1)$ its boundaries (may be the same). We can eliminate
%$O \cap \lal*$ from $\lal*$ to obtain another smaller sublamination 
%- but this contradicts minimality of $\lal*$.
%So this option cannot occur.
%
%The leaves are in this $I$-bundle and since the $I$-bundle
%is not compact the leaves limit somewhere in the manifold
%(not themselves). This implies that $\lal* - O$ is non empty.
%Then $\lal* - O \cap \lal*$ is a nontrivial sublamination.
%This is a contradiction.
%
%This finishes the proof of the lemma.
%%\end{proof}

\vskip .1in
\noindent
{\underline {Continuation of the proof of proposition \ref{as1}}}

%Regardless of whether $\fol$ is $\rrrr$-covered or not
%we get the sequences of bad wedges converging to directions
%in the limit leaves. If $F$ is such a leaf with
%limit direction associated to ideal point $a$ then for
%any closed interval $I$ in $\pin F - \{ a \}$ there
%is a ray $l$ of $\wlsf$ with subarcs $l_i$ which limit
%exactly on $I$.

%We continue the proof of the proposition, assuming that $\fol$
%is not $\rrrr$-covered. 
By the previous lemma
 we know that limit directions of
bad wedges are unique and they vary continuously in leaves
of $\widetilde {\ccl}$.
These unique directions are distinguished in their
respective leaves.

We first show that any complementary region of $\ccl$
(if any)
is an $I$-bundle with a product flow.
%and use the previous proof to obtain a contradiction.
%Let $A$ be a complementary region of $\lal*$ and let $B$
%be a boundary leaf of $A$.


Lift to a double cover if necessary to assume that $M$
is orientable. Assume this is the original foliation
$\fol$, flow $\Phi$, etc..
Let $Z$ be a leaf of $\widetilde {\cal L}$. Since $Z$
has a distinguished ideal point, then the fundamental
group of $\pi(Z)$ can be at most ${\bf Z}$. Since there
is a transverse flow and $M$ is orientable this implies
that $\pi(Z)$ is either a plane or an annulus.

Let $U$ be a complementary region of $\ccl$ with
boundary leaves $R_1, R_2, R_3$, etc..
As explained before the completion of $U$ has
a compact thick part and the non compact 
arms which are in thin, $I$-bundle
regions. 
Suppose first that $R_1$ is a plane.
There is a big disk $D$ so that $R_1 - D$ is contained
in the thin arms and flows across $U$ to another boundary
components of $U$. By connectedness it flows into a single
boundary component $R_2$ of $U$. Then $\partial D$ flows
into a curve $\gamma$ in $R_2$ which is null homotopic
in $M$. 
The flow segments in $M$ produce an annulus
$C$ in the completion of $U$. Since $\fol$ is Reebless
then $\gamma$ bounds a disk $D'$ in $R_2$ and so $R_2$
is a plane. The union $D \cup C \cup D'$ is an
embedded sphere in $M$ which bounds a ball $B$.
%As explained before 
Since orbits of $\wwp$ are properly embedded in $\mi$,
it follows that 
the flow has to a product flow
in $B$ as well. 
This shows that flow is a product
in the completion of $U$.

Suppose now that each $R_i$ is an annulus. Let
$F$ be a lift of $R_1$ to $\mi$ with $F$ in the boundary 
of a component $\widetilde U$ of $\pi^{-1}(U)$.
In $R_1$ there are two disjoint open annuli $A_1, A_2$ contained
in the thin arms so that $B = R_1 - (A_1 \cup A_2)$
is a closed annulus in the core.
Then $A_1, A_2$ flow into two annuli leaves
$R_2, R_3$ in the boundary of $U$.
Lifting to $F = \widetilde R_1$ we see leaves
of $\wlsf$ limiting in an interval of $\pin F$ with very
small complement (near the distinguished ideal
point of $F$). This implies they will have
points in the lifts $\widetilde A_1, \widetilde A_2$
of $A_1, A_2$ to $F$.
This shows that $\widetilde A_1, \widetilde A_2$
are in the same leaf of $\fn$.
This implies that $R_2 = R_3$. 
In the same way a half of the infinite strip $\widetilde B$
flows into $\widetilde R_2$. Since $B$ is compact,
then all of $B$ flows into $R_2$.
%The flow segments from
%of $\partial A_1, \partial A_3$ to $R_2$ produce
%two annuli $C_1, C_2$ in the completion of $U$.
%The boundaries of $C_1, C_2$ in $R_2$ are freely
%homotopic since $R_2$ is an annulus, hence they
%jointly bound an annulus $C_3$. Then $B \cup C_1 
%\cup C_2 \cup C_3$ is an embedded $2$-torus in
%the closure of $U$. As $M$ is atoroidal
%then this torus either bounds a solid torus
%or has to be contained in a $3$-ball. The second
%option is disallowed since the annulus $B$ is
%$\pi_1$-injective in $M$.
%It follows that the $2$-torus bounds
%a solid torus $V$,
%which must be contained
%in $U$. 
This implies that the region $U$ is an $I$-bundle.
It is also easy to show that the flow is a product
in this $I$-bundle.

This implies that we can collapse this complementary region
along flow lines to completely eliminate it.
This is because even in the universal cover we are 
eliminating product regions of the flow and the
asymptotic behavior is still preserved in the remaining
regions.
This can be done to all complementary regions and
therefore we can assume there are no complementary
regions, that is $\ccl = \fol$ or that $\fol$ is minimal.


%Suppose now that $\fol$ is not $\rrrr$-covered.
Let $F_1, F_2$ be leaves of $\fn$ which are not
separated from each other. Consider leaves $F$ of 
$\fn$ which are very close to points in both $F_1$
and $F_2$. As stated in the conclusion in the beginning
of the proof of this theorem,
there is a wedge of $F$ which flows into $F_1$
and similarly for $F_2$. Hence there are half planes
$E_1, E_2$ of $F$ which flow into $F_1, F_2$.
As $F_1, F_2$ are not separated this implies that
$E_1, E_2$ are disjoint. Fix a point $w$ in $F$ and
a big enough radius $r$ so that the disk $D$ of radius
$r$ around $w$ intersects both $E_1, E_2$.
Again as seen in the conclusion above
there is an arc $l$ in a leaf of $\wlsf$
so that both endpoints of $l$ are outside $D$ and in $E_1$
and so that $l$ is entirely outside $D$ and as seen
from $p$ the visual measure of $l$ is almost $2 \pi$.
This implies that $l$ intersects $E_2$. Since the
endpoints of $l$ are in $E_1$, which flows to $F_1$,
then proposition \ref{conn} implies that the whole arc $l$ flows
into $F_1$. The points of $l$ in $E_2$ will also flow
to $F_2$. This is a contradiction.

This contradiction 
%shows that $\fol$ has to be $\rrrr$-covered
finishes the proof of proposition \ref{as1}
\end{proof}

Next we analyse the $\rrrr$-covered situation which has interest
on its own:

\begin{theorem}{}{}
Let $\fol$ be an $\rrrr$-covered foliation and $\Phi$ be a pseudo-Anosov flow
almost transverse to $\fol$. Then $\Phi$ is actually transverse to $\fol$.
In addition for any leaf $F$ of $\fn$ and for any ray $l$ in $\wlsf$
it converges to a unique ideal point in $\pin F$.
The limit usually depends on $l$.
\label{as2}
\end{theorem}

\begin{proof}{}
If $\Phi$ is not transverse to $\fol$, let 
$\Phi^*$ be an almost 
pseudo-Anosov flow which is transverse to $\fol$ and
is a blow up of $\Phi$. 
Notice this is not the same notation as in proposition \ref{as1}
$-$ here we prove $\Phi$ is equal to $\Phi^*$.
There is flow annulus $A$ of $\Phi^*$
with closed orbits $\gamma_1, \gamma_2$ in
the boundary, so that $A$ blows down to a single orbit of $\Phi$.

The foliation induced by $\fol$ in $A$ 
has leaves which spiral to at least one
boundary component $-$ which they do not intersect.
Lifting this picture to the universal cover one
obtains an orbit of $\wwp^*$ which does not intersect
every leaf of $\fn$. This means that the flow $\wwp^*$
is not {\em regulating} for $\fn$ \
\cite{Th6, Th7}. 
We also say that $\Phi^*$ does not regulate $\fol$.
In \cite{Fe9} we analysed a similar situation and proved 
the following: if $\Upsilon$ is a pseudo-Anosov flow transverse
to an $\rrrr$-covered foliation and $\Upsilon$ is not regulating,
then $\Upsilon$ is an $\rrrr$-covered
Anosov flow.
The same arguments work with an almost pseudo-Anosov flow
transverse to an $\rrrr$-covered foliation.
This shows that $\Phi^*$ is an $\rrrr$-covered Anosov flow
and has no (topological) singularities.
In particular $\Phi^*$ is equal to $\Phi$, that is
the original flow is already transverse to $\fol$.
This proves the first assertion of the theorem.

%The same article shows that if $\Phi$ is not regulating, then the intersection
%of $\fs$ and $\fol$ is always a quasigeodesic (verify this part????).

%From now on, we assume that $\Phi$ is 
%regulating for $\fol$. 
Assume by way of contradiction that there
is $L'$ in $\widetilde {\Lambda}^s$ and $l$ in 
$\widetilde {\Lambda}^s_{L'}$ which 
does not converge to a single point in $\pin L'$.
As in the proof of theorem \ref{as1} we construct a minimal
sublamination $\ccl$ of $\fol$ such that:
for every $L$ in $\widetilde {\ccl}$ there is
an ideal point $u$ in $\pin L$ so that for every closed
segment $J$ in $\pin L - \{ u \}$ there is a ray
$l$ of $\widetilde {\Lambda}^s_L$ which has subsegments
limiting to $J$..
As shown in the proof of theorem \ref{as1}, 
$\ccl$ cannot be
a compact leaf.

%In fact if $\fol$ has a compact leaf then every minimal set is a compact
%leaf: use Brittenham's result \cite{Br1}
%to obtain the result on a manifold with boundary.

%So we can assume that 
%Hence $\ccl$ does not have compact leaves.
%Under these conditions we proved in \cite{Fe4} that
%there is a unique minimal set of $\fol$, which must
%be $\ccl$.
%Also the complementary regions of $\ccl$ are $I$-bundles
%where the flow is a product.
%Hence we can collapse all complementary regions of $\ccl$
%to obtain a new foliation which is minimal, but still transverse to the
%flow. The asymptotic properties of are the same, so we assume from
%now on that this is the original foliation, that is that $\fol$ is minimal.

%Let then $l$ in $\widetilde g$ and $l$ ray of $\wls_l$ with
%$l$ limiting. As before produce $p_i$ in $L$ and $g_i(p_i)$
%converging to $p_0$ and $L*$ and every leaf?????

%\vskip .1in
%\noindent {\underline {Claim}} - If $u \in \pin F$ is an ideal point
%of shrinking behavior then the same occurs in every other leaf.
%
%For any $u_1, u_2$ sufficiently near $u$ there is a single small
%intervalin $\pin F$ connecting $u_1, u_2$. If $S$ is any other
%leaf there is a homeomorphism $\pin F$ to $\pin S$.
%For any $S'$ between $F$ and $S$, there is a contracting
%direction in the middle of $S$. Now
%cover $I$ by these intervals. Only finitely many to go from $\pin F$ to
%$\pin S$.  So the limiting behavior comes through the intersection of these.
%Hence through $u_1, u_2$. Since $u_1, u_2$ are arbitrary, the whole picture goes
%through.
%
%\vskip .1in
%\noindent {\underline {Case 1}} - There are $a, b$ distinct in $\uu$
%which are ideal points of shrinking neighborhoods.
%
%
%
%
%
%Then there is a covering translation $g$ leaving invariant a leaf $F \in \wl$
%with ideal points which are not of $a, b$. 
%Suppose $b$ is not ideal point. COnsider the sequence of points
%associated to $b$ in $L$, where we assume that they
%are not separated by the ideal points of $g$.
%Choose one subarc intersection with axis which is
%very far away both endpoints and does not intersect
%$\cal A$ in the interior and the distance between intersections
%is less than the translation length of $\cal A$. 
%Let $e_1, e_2$ be the endpoints of this arc.
%Then there is an integer $n$ with $g^n(a')$ in the open
%arc $(e_1, e_2)$. But then $g^n(e_2)$ is on the other side
%and $g^n(a)$ intersects $a$ transversely which
%is a contradiction, see fig 4?????
%\vskip 1.5in
%\centerline{figure 4, intersections being forced}


Suppose first that every leaf of $\fol$ is a plane.
Then Rosenberg \cite{Ros} proved that $M$ is the $3$-dimensional
torus $T^3$. 
This manifold is a Seifert fibered space. In this case
Brittenham \cite{Br1} proved that an essential lamination
is isotopic to one which is either vertical (a union
of Seifert fibers) or horizontal (transverse to the 
fibers). So after isotopy assume $\ccl$ has one
of these types. If $\ccl$ has a vertical leaf $B$, then
geometrically it is a product of the reals with the circle.
Hence it is an Euclidean  leaf and in the universal
cover it has polynomial growth of area.
If $\ccl$ has
a horizontal leaf $B$, then
because the fibration is a product,
there is a projection to a $T^2$ fiber, which distorts
distances by a bounded amount. Again the same growth properties
hold.  But the leaves of $\fol$ are hyperbolic, which is 
a contradiction. We conclude that $M$ cannot be $T^3$.





%\vskip .1in
%noindent
%\underline {Case 2}} - In this case $a$ in $\uu$ is uniquely determined
%nd it is invariant under the group action.




%\blankfig{cros}{1.1}{Forcing the crossing of leaves.}

Let then $F$ in $\widetilde {\ccl}$ with $\pi(F)$ not simply 
connected. Let $g$ in $\pi_1(M)$ non trivial with $g(F) = F$
and $\xi$ be the axis of $g$ in $F$. At least one ideal point
of $\xi$, call it $u$, is not the direction of a fixed 
limit of bad wedges. Then as explained before there is a 
ray $l$ of $\wlsf$ and segments $l_i$ 
of $l$, bounded by $a_i, b_i$ both points
in $\xi$, so that $l_i$ escapes compact sets
and converges to a non trivial segment in $\pin F$.
We may assume that $l_i \cap \xi = \{ a_i, b_i \}$ and also
that all $l_i$ are in the same side of $\xi$.
Let $e_0$ be the translation length of $g$ in $F$.

If the distance from $a_i$ to $b_i$ along $\xi$ is bigger than $e_0$ then
this produces a contradiction as follows:
There is an integer $n$ so that $g^n(a_i)$ is  in the 
open segment $(a_i,b_i)$ of $\xi$ and
and $g^n(b_i)$ is outside of the closed segment $[a_i,b_i]$.
Since the arc $l_i$ only intersects $\xi$ in $a_i, b_i$, then 
$l_i$, together with $[a_i,b_i]$
bounds a closed disk in $F$ and $g^n(a_i)$ is in $(a_i,b_i)$.
But $g^n(b_i)$ is outside and $g^n(l_i)$ is also on this side
of $\xi$, so this produces
a transverse self intersection of $\wlsf$.
If $g^n(l_i)$ is contained in the leaf $v$ which contains $l_i$,
then $g^n(v) = v$ and this produces infinitely many
singularities in $v$, which is impossible. Hence $g^n(l_i)$ is
not in $v$ and the transverse intersection is impossible.
The same arguments deal with the case that $l_i$ intersects
$\xi$ in other points besides $a_i, b_i$.

%First i%f $l_i$ intersects $a$ in the
%other end we can map in the middle and again get a
%self intersection.

We conclude that the distance in $\xi$ from $a_i$ to $b_i$ is bounded.
%But $d_{l_i}(b_i,c_i)$ is not. 
Up to subsequence we may assume there are integers $n_i$
so that 
$g^{n_i}(a_i)$ converges to $a_0$ and $g^{n_i}(b_i)$ converges to $b_0$,
both limits in $\xi$ of course.
Since the lengths of $g^{n_i}(l_i)$ are converging to infinity,
it follows that $a_0, b_0$ are not in the same leaf of $\wlsf$.
By proposition \ref{conn} it follows that $a_0, b_0$ are not
in the same leaf of $\wls$. But for each $i$, the pair
of points $g^{n_i}(a_i), g^{n_i}(b_i)$ is in the same leaf
of $\wls$. This implies that the leaf space of $\wls$ is
not Hausdorff.

First of all this implies that $\Phi$ is regulating for $\fol$,
for otherwise the aforementioned result from \cite{Fe9}
shows that $\Phi$ is an $\rrrr$-covered Anosov flow $-$ in
particular $\wls$ has Hausdorff leaf space.
Also by theorem \ref{theb} the fact that $\wls$ has non Hausdorff
leaf space implies that there are closed orbits $\alpha, \beta$
of $\Phi$ so that $\alpha$ is freely homotopic to the inverse
of $\beta$.
Let $h$ be a covering translation associated to $\alpha$ and
$\widetilde \alpha$, $\widetilde \beta$ lifts of $\alpha, \beta$
to $\mi$ which are left invariant by $h$. Without loss of generality
assume that $h$ acts in $\widetilde \alpha$ sending points forwards.
As $\alpha \cong \beta^{-1}$ this implies that $h$ acts
on $\widetilde \beta$ taking points backwards. But since
both of them intersects all leaves of $\fn$ (by the regulating
property) then
as seen from $\widetilde \alpha$ the translation $h$ acts
increasingly in the leaf space of $\fn$, with opposite
behavior when considering $\widetilde \beta$.
This is a contradiction, which shows that this cannot happen.
This finishes the proof of theorem \ref{as2}.
\end{proof}


%Because the lengths of the $l_i$ between $b_i$ and $c_i$ 
%grow without bound, then $b_0, c_0$ are not in the same
%leaf. This implies that the leaf space of
%$\wl$ is not Hausdorff. This implies that there are closed
%orbits $\alpha, \beta$ of $\Phi$ in $M$ which are freely
%homotopic to the inverse of each other. But since
%$\Phi$ is regulating, this is impossible. This finishes
%the proof of the theorem.

Proposition \ref{as1} and theorem \ref{as2} imply the following:

\begin{corollary}{}{}
Let $\Phi$ be an almost pseudo-Anosov flow transverse to
a foliation $\fol$ with hyperbolic leaves in $M^3$ closed.
For any leaf $L$ of $\fn$ and any ray $l$ of $\wlsf$
or $\wluf$, then $l$ converges to a single point in $\pin L$.
\end{corollary}



\vskip .1in
\noindent
{\bf {Remark}} - Group invariance and compactness of $M$ are both
essential here.  For example start with a nicely behaved singular foliation
of $\hh$, so that all rays converge. It could be a foliation by geodesics
or for instance the lift of the stable singular foliation associated
to a suspension. 
Fix a base point $p$. Now rotate the leaves at a distance $d$ of $p$ by
an angle $d$. In this situation all rays limit in all points of $\pin L$,
in fact they spiral indefinitely into it.
Another operation is to fix a ray through $p$ and then distort the rest more
and more one way and the other way. Here we have the leaves getting
closer and closer to segments in $\pin F$ which 
are complementary to the ideal point associated to the
ray.




\section{Properties of leaves of $\wlsf, \wluf$ and their ideal points}

In this section $\Phi$ is an almost pseudo-Anosov flow
transverse to a foliation $\fol$. 
As in the previous section
there is no restriction on $M$ here.
In the previous section we proved that
for any ray $r$ of a leaf of $\wlsf$ or $\wluf$, then
it has a unique ideal point in $\pin F$. The notation for this ideal
point will be $r_{\infty}$.
We now analyse further properties of leaves of $\wlsf$ and their
ideal points.
Analogous results hold for $\wluf$.

First we want to show that if $E$ is a fixed
leaf of $\wls$ (or $\wlu$)
then the ideal points in $\pin F$ of rays of $E \cap F$
vary continuously with $F$.
In order to do that we first
put a topology on the union of ideal boundaries of an interval
of leaves. Let $p$ in $F$ leaf of $\fn$ and $\tau$ a transversal
to $\fn$ with $p$ in the interior.
For any $L$ in $\fn$ intersecting $\tau$, the ideal
boundary is in 1-1 correspondence with the unit
tangent bundle to $L$ at $\tau \cap L$:
ideal points correspond to rays in $L$ starting at $L \cap \tau$.
This is a homeomorphism.
This puts a topology in 

$$ {\cal A}  \ \ = \ \ 
\cup \ \{ \pin L \ \ | \ \ L \cap \tau \not = \emptyset \}$$

\noindent
making it into an annulus homeomorphic to 
$\cup  \ \{ T^1_q \fn, q \in \tau \}$ as a subspace
of the unit tangent bundle of $M$.
This topology in ${\cal A} $ is independent of the choice of transversal
$\tau$. The following definition/result is proved
in \cite{Fe7} or \cite{Cal1}.

\begin{define}{(markers)}{}
Given a foliation $\fol$ by hyperbolic leaves
of $M^3$ closed, then  there is $\epsilon > 0$ so
that:
Let $v$ be a geodesic ray in a leaf $F$ so that 
it is associated to a contracting (or $\epsilon$ non
expanding) direction of $F$.
For any leaf $L$ sufficiently near $F$, then
all the points of $v$ flow into $L$ and define
a curve denoted by $v_L$. 
Then $v_L$ has a unique ideal point denoted by $a_L$. The
union $m$ of the $a_L$ is called a marker and is
a subset of ${\cal A}  = \cup \ \{ \pin L \}$. 
Then $m$ is an embedded curve in ${\cal A} $ in the topology
defined above.
\end{define}


In addition the markers are dense in ${\cal A} $ in the following sense:
Let $z$ in $\pin F$ and $a_i, b_i$ in $\pin F$ which are 
in markers associated to contracting (non expanding) directions
on a fixed side of $F$. Suppose that the sequence of open 
intervals
$(a_i, b_i)$ in $\pin F$ contains $z$ and converges
to $z$ as $i$ converges to infinity.
Let $\alpha_i, \beta_i$ be the markers in that
side of $\pin F$ containing $a_i, b_i$ respectively.
Let $L_i$ in $\fn$ be a sequence of leaves converging
to $F$ and on that side of $F$ so that $\pin L_i$ intersects
both $\alpha_i$ and $\beta_i$.
In 
the annulus ${\cal A} $ of circles at infinity, consider the
rectangle $R_i$ bounded by $(a_i, b_i)$ in $\pin F$,
the parts of $\alpha_i, \beta_i$ between $\pin F$ and
$\pin L_i$ and the small segment in $\pin L_i$ bounded
by \ $\pin L_i \cap \alpha_i$ \ and \ $\pin L_i \cap \beta_i$.
Then the sets $R_i$ converge to $z$  as $i$ converges
to infinity. This is proved in \cite{Fe7}.


From now on the $\epsilon$ is chosen small enough
to also satisfy
the conclusions of the definition above and also that
any set in $\mi$ of diameter less than $10 \epsilon$
is in a product box of $\fn$ and $\wwp$.
Given a curve $\zeta$ in a leaf $F$ with starting
point $p$ and limiting on a unique point $q$ in $\pin F$,
let $\zeta^*$ denote the geodesic ray of $F$ with
same starting and ideal points.






\begin{lemma}{}{}
Let $E$ be a leaf of $\wls$ and $p$ the starting point of the ray $r$ of 
$E \cap F$.
Assume that $r$ does not have any singularity.
For any $L$ near $F$, then $E \cap L$ has a ray
$r_L$ which is near $r$. The ideal points
of $r_L$ in $\pin L$ vary continuously with $L$
in the topology of ${\cal A}$ defined above.
\label{cont}
\end{lemma}

\begin{proof}{}
We do the proof for say the positive side of $F$.
We consider $r$ without singularity or else we would
have to check the 2 exterior rays in $\wlsf$ emanating from
$p$. We can always get a subray of $r$ which has
no singularities.

Let $u = r_{\infty}$.
 Choose 
contracting (or $\epsilon$ non expanding) directions  
in both sides of $u$, with ideal points very close to $u$.
Let them be defined by geodesic rays $r_0, r_1$ starting at $p$.
There is $\tau$ a small flow segment starting at $p$ and
 in that side of $F$  so that for any $L$ intersecting
$\tau$, then $L$ is asymptotic to $F$ along the
$r_0, r_1$ rays, or at least always $\leq \epsilon$ from $F$.
Hence $r_0, r_1$ flow along $\wwp$ to $L$. Let 
$s_0, s_1$ be the flow images in $L$. 
The $\epsilon$ is also chosen small enough so that 
$s_0, s_1$ have geodesic curvature very small
(this $\epsilon$ depends only on $M$ and $\fol$).
In particular the curves $s_0, s_1$ 
are a small bounded distance (depending only on $\epsilon$) 
from
the corresponding geodesic arcs $s^*_0, s^*_1$.
Let the 
ideal points of $s_0, s_1$  in $\pin L$
be denoted by  $v_0, v_1$
and let $J_L$ be the small closed interval
in $\pin L$ bounded by $v_0, v_1$.
Then $v_0, v_1$ 
are in the markers
associated to $r_0, r_1$ respectively and so they vary continuously
with $L$.


\begin{figure}
\centeredepsfbox{gm6.eps}
\caption{
Leaf in wedge defined by markers.}
\label{bet}
\end{figure}



Consider $\xi = E \cap L$ and the rays $l$ of
$\xi$ starting at $\tau \cap L$ and
containing some points which flow back to points
in $r$. It may be that $\xi$ has singularities $-$
even if $r$ does not $-$ but there are only
finitely many such rays.
We want to prove that the ideal point of any such 
is in $J_L$.
As the rectangles $R_i$ defined above converge to 
$u$ in 
${\cal A}$ this will prove the continuity property of the lemma.


Choose $d > 0$ so that outside of a disk $D$
of radius $d$ in $F$, then
$r$ is in the small wedge $W$ of $F$ defined by $r_0, r_1$,
see fig. \ref{bet}.
Choose $\tau$ small enough so that if $L$ intersects $\tau$, then
the entire disk $D$ is
$\epsilon$ near $L$.
Let $V$ be the closure in $F$ of $W - D$.
The boundary $\partial V$ consists of subrays 
of $r_0, r_1$ and an arc in $\partial D$. Therefore
all points in $\partial V$ are less than $\epsilon$
from $L$ and flow to $L$ under $\wwp$ with image
a curve $\gamma$. This curve contains subrays of 
$s_0, s_1$ and it is properly embedded in $L$.
Points of $F$ near $\partial V$ also flow to $L$ so there
is a unique component $U$ of $L - \gamma$ which has
some points flowing back to points in $V$.
We want to show that the ray $l$ is eventually
contained in $U$.

Let $r_{init}$ be the subarc of $r$ between $p$ and
the last point $c_0$ of $r$ in $D$.
As $p$ and $c_0$ flow into $L$, then proposition \ref{conn}
shows that the entire arc $r_{init}$ flows into $L$
and let $\delta$ be its image in $L$.
As $r$ is singularity free, then so is $\delta$ and
hence $\delta$ is contained in any ray $l$ of $E \cap L$
in that direction.
After $c_0$ the curve $r$ enters $V$ and so $l$ must
enter $U$ after $\delta$.
If after that the ray $l$ exits $U$ then it must
cross $\partial U = \gamma$ in some point, call it $c_1$.
But $c_1$ flows back to $F$ and one can apply
proposition \ref{conn} again in the backwards direction
to show that $c_1$ has to flow to a point in $r$.
This contradicts the choice of $c_0$.

This shows that $l$ is eventually entirely contained
in $U$ and therefore $l_{\infty}$ is a point
in $J_L$. This shows the continuity property 
as desired and finishes the proof of the lemma.
\end{proof}

Now we have a property which will be crucial to a lot of our analysis.

\begin{proposition}{}{}
Suppose that $\fol$ is not topologically conjugate to the
stable foliation of a suspension Anosov flow.
Then the set of ideal points of rays of $\wlsf$ is dense in
$\pin F$.
\end{proposition}

\begin{proof}{}
Suppose that there is $F$ in $\fn$ so that the set of
ideal points in $\wlsf$ is not dense in $\pin F$.
Let $J$ be an open interval in $\pin F$ free of such
ideal points. Choose $p_i$ in $F$, $p_i$ converging
to a point in $J$. 
The visual angle of $J$ as seen from $p_i$ converges
to $2 \pi$, so the complementary wedge $W_i$
with corner  $p_i$ has angle which converges to
zero. Up to subsequence assume that 
$g_i(p_i)$ converges to $p_0$ in a leaf $L$ of $\fn$
and the small wedges $g_i(W_i)$ converge to a geodesic
ray $s$ in $L$ with ideal point $z$.

\vskip .15in
\noindent
{\underline {Claim }} $-$ In $L$ all the rays of $\wlsl$ converge
to $z$.

%This is very similar to previous arguments. 
Suppose there is
$x$ different from $z$ which is an ideal point of a ray
$r$ in $\wlsl$. Then $r$ is contained in $\ws(c_0)$ for
some $c_0$ in $\mi$  and
for $g_i(F)$ sufficiently near $L$ then $\ws(c_0)$ intersects
$g_i(F)$. Any 
ray of $\ws(c_0) \cap g_i(F)$ which is near $r$ will have ideal point
near $x$ in the topology of corresponding annulus ${\cal A}$
of ideal circles near $\pin L$. This is a consequence of the previous
lemma.
But $g_i(W_i)$ converges to $r$ in this topology of ${\cal A}$,
so the sets $g_i(\pin F - J)$ converge to $z$ in ${\cal A}$.
There are no ideal points of leaves of 
$\widetilde {\Lambda}^s_{g_i(F)}$ in $g_i(J)$.
This contradicts the fact that the ideal points above are
very near $x$ and proves the claim.
\vskip .1in

The proof of the proposition
is similar to that of theorem \ref{as1}.
As in that theorem consider the set of possible limits
$g_i(p_i)$ as above. This projects to a lamination in $M$
and let ${\cal L}$ be a minimal sublamination.
The claim shows that each leaf of $\wcl$ has a distinguished
ideal point towards which all rays of $\wlsl$ converge.
The arguments in the claim also prove that if $\tau$ is
a transversal to $\fn$, then the ideal points of
leaves of $\wcl$ intersecting $\tau$ vary continuously
in the corresponding ideal annulus. Because of the
distinguished ideal point property, then each leaf of
${\cal L}$ has fundamental group at most ${\bf Z}$.
If needed lift to a double cover so that all leaves
of $\fol$ are orientable. Hence a leaf of ${\cal L}$ is
either a plane or an annulus.


Consider a complementary component $U$ of ${\cal L}$
and a boundary leaf $A$ of $U$. If $A$ is a plane then
as in the proof of theorem \ref{as1},
the region $U$ is an $I$-bundle over $A$ and the flow
$\Phi$ is a product in $U$. This region can be
collapsed away.

Suppose now that $A$ is an annulus.
Assume that flow lines through $A$ flow into $U$.
Again we want to show that $U$ is a product region.
As in the proof of theorem \ref{as1} let $A_1, A_2$ be
two noncompact, disjoint annuli in $A$ with $A - (A_1 \cup A_2)$
a compact annulus and $A_1, A_2$ contained in the 
thin, $I$-bundle region. Then $A_1, A_2$ flow entirely
into leaves $B$ and $C$ in $\partial U$. 
%If $B = C$ then
%as in theorem \ref{as1}, the region $U$ is a product with
%product flow.
Suppose first that $B, C$ are different. Lift to the
universal cover to produce lifts $\widetilde U, \widetilde
A, \widetilde A_1, \widetilde A_2, \widetilde B, \widetilde C$.
Then $\widetilde A_1, \widetilde A_2$ are disjoint half
planes of $\widetilde A$ which flow positively respectively
into $\widetilde B$ and $\widetilde C$. 
Let $g$ be the generator of the isotropy group of $\widetilde A$,
which has fixed points in $z, x$ where $z$ is the
distinguished ideal point in $\widetilde A$.
The argument will show there is a leaf in
$\widetilde {\Lambda}^s_{\widetilde A}$ which also has
ideal point in $x$, contradiction.

From a point in 
$\widetilde A_1$ draw a geodesic segment of 
$\widetilde A$ to a point
in $\widetilde A_2$. Let $p$ be the first point
of this segment which does not flow positively into 
$\widetilde B$.
Then $\Theta(p)$ is in the boundary of $\Theta(\widetilde B)$.
Also  points in the segment near $p$ flow to $\widetilde B$
in positive time, hence there is a slice leaf $l$ of
$\oos(\Theta(p))$  which is in the boundary of $\Theta(\widetilde B)$.
Notice that every point in $l$ is a limit of points in $\Theta(B)$ on
that side.
The set $(l \times \rrrr)$ intersects $\widetilde A$ in at least
$p$: if
$l$ is contained in $\Theta(\widetilde A)$ then it 
generates a properly embedded copy of the reals
in a leaf $s$ of $\widetilde {\Lambda}^s_{\widetilde A}$
otherwise the part that is contained  in $\Theta(\widetilde A)$
also does. 
Every point of $s$ is a limit of points  that flow
%of $\widetilde A_1$ on
%that side of $s$, so that these near points flow 
positively into $\widetilde B$. Therefore no point in $s$
can flow positively in $\widetilde C$ or else we would have
points flowing both in $\widetilde B$ and $\widetilde C$.


This shows that the leaf $s$ of $\widetilde {\Lambda}^s_{\widetilde A}$
is a bounded distance from the axis $r$ of $g$.
Iterate $s$ by powers of $g$ acting with $z$ as an expanding
fixed point.
The iterates $g^n(s)$ with $n > 0$ are all distinct. 
Either they are all nested or they are disjoint.
If they are not nested since they all have to be in a bounded
distance neighborhood of the axis of $g$ and have both
endpoints in $z$, then eventually they will have two 
points which are far along the leaf, but close in $\widetilde A$.
By Euler characteristic reasons, this would force
a center or one prong singularity, which is impossible.
Hence they are nested, increasing and 
they limit to a leaf of $\widetilde {\Lambda}^s_{\widetilde A}$
which has ideal limit points in $z$ and $x$. This is a contradiction.
This shows that $B = C$.
In fact the same arguments show that all of the points
in $A$ flow into $B$, since that happens for the complement
of a compact annulus in $A$ and then the arguments above 
apply here.
Hence $U$ is a product region.
Therefore
we can collapse $\fol$
to a minimal foliation. 

%There is a subannulus which is the core. The rest is in the interstitial
%bundle. An annulus flows to a leaf $A$. From $B$ we consider all the
%flow lines to $A$. It contains a half plane and is disjoint from the
%other half plane.
%If the other side also maps to $A$ then if everything maps to
%$A$ we are done. Else it does not map to $A$. The boundary of the set
%where it does not map to $A$. If a leaf of $\wls_B$ does not
%map to $A$ entirely it has to stop somewhere. It 
%cannot stop in $A$ going down, since $B$ is below it.
%Then $B$ would escape. So it escapes going up,
%see fig. \ref{esc}.
%
%\blankfig{esc}{1.}{Escaping of nearby leaves}
%
%Since stable this only happens if the $B$ intersects the orbit
%$C$ but not $A$. Then $A$ escapes up and there is a stable leaf
%in $\partial \Theta(A)$. This intersects $B$ $-$ so it is
%entirely contained in $B$ not going to $A$ $-$ every point
%is in $\partial \Theta(A)$ so it is in the middle strip following
%the geodesic. So we produce a leaf $l$ of $\wls_B$ entirely
%in  a bounded neighborhood of a geodesic. Iterate by $g$ and
%take a limit $-$ they have to be nested, otherwise violate
%transversality $-$ do not connect the same leaf. So the
%boundary of these is a  leaf with ideal points $p$ and $q$.
%But since nested this clearly cannot happen.
%Also because outer leaf is contained in the region, we conclude
%that all complementary regions are products. 




\vskip .1in
As in theorem \ref{as1}
we can then show that $\fol$ is $\rrrr$-covered. Suppose this
is not the case and let $F_1, F_2$ be non separated leaves.
Let $L_i$ in $\fn$ leaves converging to both $F_1, F_2$.
Let $u_1, u_2$ be the distinguished ideal points
in $\pin F_1, \ \pin F_2$  respectively.
Let $a_1, b_1$ be points in $\pin F_1$ very near 
$u_1$ and on opposite sides of $u_1$ and which are
in markers associated to contracting or $\epsilon$
non expanding directions in $F_1$ associated to the $L_i$
side. 
Let $r_1$ be the geodesic in $F_1$ with ideal
points $a_1, b_1$.
Similarly for $F_2$ producing $a_2, b_2, r_2$.
For $i$ big enough $L_i$ is at most $\epsilon$ far
from all points in $r_1, r_2$. Therefore $r_1$ flows (by $\wwp$)
into a curve $s_1$ in $L_i$ and $r_2$ flows into
$s_2$.  This implies that $s_1, s_2$ are disjoint
in $L_i$.
Also $s_1$ has ideal points $a'_1, b'_1$
which are in markers containing $a_1, b_1$ respectively
(this is using a transversal to $\fn$ through a point
in $F_1$). Similarly $s_2$ has ideal points $a'_2, b'_2$
in markers containing $a_2, b_2$
(using transversal to $\fn$ through a point in $F_2$).
As $s_1, s_2$ are disjoint then $a'_1, b'_1$ do not
link $a'_2, b'_2$ in $\pin L_i$, see fig. \ref{link}.




\begin{figure}
\centeredepsfbox{gm7.eps}
\caption{
Pushing ideal points near.}
\label{link}
\end{figure}



The ideal point $a'_1$ cannot be in a marker to $\pin F_1$
and to $\pin F_2$ at the same time since they are non
separated leaves. Hence the points $a'_1, b'_1,
a'_2, b'_2$ are all distinct.
Let $J_1$ be the interval of $\pin L_i$ bounded by
$a'_1, b'_1$ and not containing the other points
and similarly define $J_2$.
For simplicity we are ommitting the dependence of $J_1, J_2$ 
on $L_i$ (or on $i$).
Now consider $E$ a leaf of $\wls$ intersecting $F_1$.
Then $E \cap F_1$ has a ray with ideal point $u_1$,
which is in the interval $(a_1,b_1)$ of $\pin F_1$.
The proof of lemma \ref{cont} shows that if
$L_i$ is close enough to $F_1$ then the ideal
points of the corresponding rays of $(E \cap L_i)$ 
have to be in $J_1$. In the same way using $F_2$ one
shows that the distinguished ideal point has to be
in $J_2$. Since $J_1, J_2$ are disjoint, this is
a contradiction. This shows that $\fol$ is
$\rrrr$-covered.






Since $\fol$ is $\rrrr$-covered then theorem \ref{as2} implies
that $\Phi$ can be chosen to be a pseudo-Anosov flow.
%In addition it is regulating, for otherwise the intersection
%of flow lines and leaves are geodesics??????.

Also as $\fol$ is $\rrrr$-covered we can choose a transversal
$\tau$ intersecting all the leaves of $\fn$. This shows that
the union of all the circles at infinity has a natural topology
making it into a cylinder ${\cal A}$. This situation of
$\rrrr$-covered foliations is carefully analysed in \cite{Fe7}.
The fundamental group of $M$ acts in ${\cal A}$ by
homeomorphisms.
The union of the distinguished
ideal points of leaves of the distinct leaves of $\fn$
is a continuous curve $\zeta$ in ${\cal A}$ which
is group invariant.

Suppose first that $\fol$ admits a holonomy invariant
transverse measure. Since $\fol$ is minimal then
the transverse measure has full support.
Under these conditions Imanishi \cite{Im} proved
that $M$ fibers over the circle
with fiber a closed surface.
In addition $\fol$ is approximated arbitrarily near
by a a fibration.
The pseudo-Anosov flow is also transverse to these
nearby fibrations and so the same situation occurs
for the fibrations: there is a global invariant
curve in the cylinder at infinity. Since now
there are compact leaves, this is impossible.

We conclude that there is no holonomy invariant
transverse measure. Therefore Thurston's theorem
shows the existence of contracting directions
and not just $\epsilon$ non expanding directions.
So the markers are associated to contracting directions.
If $\zeta$ intersects a marker $m$, that corresponds
to a direction in a leaf of $\fn$ which is contracting.
Under the flow $\wwp$ this
gets reflected in the contracted leaves nearby, that
is the marker is contained in $\zeta$.
Since $\fol$ is minimal and $\zeta$ is $\pi_1(M)$ invariant,
this shows that the entire curve $\zeta$ is
a marker associated to  contracting directions.
The results from \cite{Fe7} apply here, in particular
lemma 3.17  through proposition 3.21 of \cite{Fe7}:
they show that no other direction in $\fn$ (outside
of $\zeta$) is a contracting direction.
%This implies that no other direction in leaves
%of $\fn$ can be a contracting direction.
By Thurston's theorem again, there would be a
holonomy invariant transverse measure, contradiction.

Therefore $\zeta$ has no contracting directions.
The same analysis of \cite{Fe7} now shows that 
%This shows that $\zeta$ cannot cross from one side
%of the marker to the other side, that is, no transversal
%intersection with markers. 
%This curve $\zeta$ cannot cross any marker why???
for any leaf $F$ in $\fn$
and every direction other than the distinguished direction,
then it is a contracting direction.
In fact it is a contracting direction with any other leaf
of the foliation.



This is a very interesting situation. Let $a_F$ be the
distinguished ideal point of $F$ leaf of $\fn$.
Consider a one dimensional 
foliation in $\mi$ whose leaves are geodesics in leaves $F$ of $\fn$ 
and which have one ideal point $a_F$.  Let $\widetilde \xi$ be the
flow which is unit speed tangent to this foliation and moves
towards the ideal point $a_F$. 

This is a flow in $\mi$. Clearly in each leaf of $\fn$, it is a
smooth flow. If $q_i$ in $L_i$ of $\fn$
converge to $q$ in $L$, then the
geodesics of $L_i$ with ideal point $a_{L_i}$ converge to the 
geodesic through $q$ in $L$ 
with ideal point $a_L$. This is 
because the ideal points $a_F$ vary continuously with $F$
and $q_i$ converges to $q$ $-$ this is the local
trivialization of the union of the circles at infinity
using the tangent bundles to a transversal.
Hence $\widetilde \xi$ varies continuously.
% and clearly
%is group equivariant (this is where the hypothesis is used).

Since $\zeta$ is group invariant, this induces a flow 
in $M$, which is tangent
to the foliation $\fol$. Clearly it is smooth along the leaves of
$\fol$ and usually just continuous in the transverse direction.

This flow is a topological Anosov flow: the stable
foliation is just the original foliation $\fol$. 
The unstable foliation: Let $p$ in leaf $L$ of $\fn$, let
$\gamma$ be the flow line of $\widetilde \xi$ through $p$.
Then $\gamma$
has positive ideal point  $a_L$ and negative ideal point
$v$.  As explained above $v$ is in a marker $m$ which is associated
to a contracting direction and so that $m$
intersects
all ideal circles. For each $F$ in $\fn$, let $m_F$ be
the intersection of $m$ and $\pin F$. Let $\gamma_F$ be
the geodesic in $F$ with ideal points $a_F$ and $m_F$.
Let $E_p$ be the union of these $\gamma_F$. Then all
orbits of $\widetilde \xi$ in $E_p$ are backwards
asymptotic by construction. By construction the $E_p$ are
either disjoint or equal as $p$ varies in $\mi$ and they
form a group invariant foliation in $\mi$. This is the
unstable foliation.
Hence $\xi$ is a topologically Anosov flow. 
Notice that in the universal cover 
every stable leaf intersects every unstable leaf
and vice versa.

By proposition \ref{susp} it follows that $\xi$ is topologically
conjugate to a suspension Anosov flow.
The foliation $\fol$ is then topologically conjugate
to the stable foliation of this flow.
This finishes the proof of this proposition.
\end{proof}

%As seen in the previous section this implies that the flow
%is topologically conjugate to a suspension Anosov flow.
%Now suppose there is a unique idealpoint. Then this induces an invariant
%vertical line. This implies that $\Gamma$ is conjugate to the stable
%foliation of a suspension Anosov flow $-$ but we will not need that.
%Now consider a covering translation $g$ with an invariant leaf
%$F$ and axis $a$ and translation length $r$. As seen before there
%are arcs $l_i$ with endpoints in $a$ and limiting more and more on
%big segments in $\pin F$. We assume the intersection of $l_i$ and
%$a$ is just two points $b_i, c_i$ in the boundary of $l_i$. 
%\vskip .1in
%
%Let $\gamma$ be a closed orbit of $\Phi$ with a lift
%$\widetilde \gamma$ to $\mi$ and $g(\widetilde \gamma)
%= \widetilde \gamma$ generator of the isotropy group of $\widetilde
%\gamma$. As explained in the proof of proposition ??? $\fol$
%is topologically conjugate to the stable foliation of
%a suspension Anosov flow. Since $g$ acts freely in
%%the leaf space of $\wl$, then it is associated to
%an element of $\pi_1(T)$, where $T$ is a fiber for the
%fibration. Therefore acts by an isometry of the hyperbolic
%structure in $M$. But the return map is a contraction, at least
%along $\ws(\widetilde \gamma) \cap F$. So we obtain
%a contradiction. This finishes the proof of the result.


\vskip .1in
\noindent
{\bf {Remark}} $-$ The hypothesis is necessary. Suppose that
$\fol$ is the stable foliation of a suspension Anosov flow,
$\xi$ so that it is transversely orientable. Perturb the flow
slightly so that flow lines are still tangent to the original
{\underline {unstable}} foliation of  $\xi$. The new flow, 
call it
$\Phi$ is transverse to $\fol$, it 
has the same unstable foliation as $\xi$ but different
stable foliation. 
The flow $\Phi$ is not regulating for $\fol$. The intersections
of leaves of $\wls$ with leaves $F$ of $\fn$ are all horocycles
with the same ideal point which is the positive ideal point
of flow lines in $F$. So the ideal points of rays of leaves of
$\wlsf$ are not dense in $\pin F$. Notice  these leaves
are not quasigeodesics in $F$ either.
This example is studied in detail in section 7 of \cite{Fe9}.
\vskip .1in

Now we want to study metric properties of  slices of 
leaves of $\wlsf$.
The best metric property such leaves could have is that they
are {\em quasigeodesic}: this means that length along the
curve is at most a bounded multiplicative distortion
of length in the leaf $F$ of $\fn$ \cite{Th1,Gr,Gh-Ha,CDP}. 
If the bound is $k$ then we say the curve is a $k$-quasigeodesic.
Since $F$ is hyperbolic this would imply that such leaves
(the non singular ones) are a bounded distance from
true geodesics. 
Very unfortunate for us, this is not true in general. 
But there are still some good properties. 
%We start
%with the case that $\wls$ has Hausdorff leaf space,
%in which case the quasigeodesics property holds.

Let $\hhs$ be the leaf space of $\wls$ and $\hhu$ be
the leaf space of $\wlu$.
Clearly since $\hhs$ may be non Hausdorff, it could
be that some $\wlsf$ does not have Hausdorff leaf space. 
This easily would imply that the slices of $\wlsf$ are
not uniformly quasigeodesic \cite{Fe1}.
This in fact occurs, see Mosher \cite{Mo1,Mo5}. Still it could be
that given a ray in $\wlsf$, it is a quasigeodesic $-$
with the quasigeodesic constant depending on the particular
ray. We are not able to prove this and we cannot conjecture
what happens in generality. But we are able to
prove a weaker property, which will be enough for our purposes.
If $r$ is a ray in a leaf of $\wlsf$, recall that 
$r^*$ is the unique
geodesic ray in $F$ with same starting point as $r$ and same
ideal point. We would like to prove that $r, r^*$ are a bounded
distance apart, but we do not know if that is true. But 
we can prove the following important property:

\begin{lemma}{}{}
There is $\delta_0 > 0$ so that for any $F$ in $\fn$ and any ray $r$ in
a leaf of $\wlsf$, then given any segment of length $\delta_0$ in
$r^*$, there is a point in this segment which is less than
$\delta_0$ from $r$ in $F$. That implies that $r^*$ is in the neighborhood
of radius $2 \delta_0$ of $r$ in $F$.
\label{boun}
\end{lemma}

\begin{proof}{}
This means that 
%$r$ is roughly near $r^*$, that is 
$r^* \subset N_{2 \delta_0}(r)$ in $F$. We do not know if the converse
holds. Suppose the lemma is not true. Then there are $F_i$ leaves
of $\fn$, $r_i$ rays of $\wlsfi$
and $p_i$ in $r^*_i$ so that $B_{2i}(p_i)$ (in $F_i$) does not
intersect $r_i$. There is one 
 side of $r^*_i$ in $F_i$ so that $r_i$ goes
around that side, see fig. \ref{aroun}, a.
Let $q_i$ inside a half
disk of $B_{2i}(p_i)$ 
with $B_i(q_i)$ tangent to $r^*_i$ and
$\partial B_{2i}(p_i)$, see fig. \ref{aroun}, a.


\begin{figure}
\centeredepsfbox{gm8.eps}
\caption{
a. Limits of points, b. Going around disks
in $F_i$,
c The picture in $L$.}
\label{aroun}
\end{figure}



As usual up to subsequence there are $g_i$ in $\pi_1(M)$ with
$g_i(q_i)$ converging to $q_0$ in $L$ leaf of $\fn$
and so that the geodesic
segments $\zeta_i$ from $g_i(q_i)$ to $g_i(p_i)$ in $F_i$
converge to a geodesic
ray $s$ in $L$. Choose two markers with 
points $u_0, u_1$ in $\pin L$ very close to $s_{\infty}$
and on opposite sides of it.
The markers are associated to the side of $L$
where the $g_i(F_i)$ are limiting to.
Let $s_0, s_1$ be the geodesic rays of $L$ starting at $q_0$ and
with ideal points $u_0, u_1$.
For $i$ big enough $g_i(F_i)$ is $\epsilon$ close to both
$s_0$ and $s_1$ and so these two rays flow (under $\wwp$)
to curves $s'_0, s'_1$ in $g_i(F_i)$. 
The ideal points $u'_0, u'_1$ of $s'_0, s'_1$ are in the markers
above.

For $i$ big enough the ray $g_i(r_i)$ has a subray which goes 
around $g_i(B_i(q_i))$ in $g_i(F_i)$ and has ideal
point in the small segment of $\pin g_i(F_i)$ defined
by $u'_0, u'_1$, see fig. \ref{aroun}, b.
Since $s'_0, s'_1$ flows back to $L$ this figure flows
back to $L$ producing a ray $l_i$ of $\wlsl$
which goes around a big disk in $L$ centered at $q_0$
and has ideal point in the small segment bounded by
$u_0, u_1$, see fig. \ref{aroun}, c.
As $i$ goes to infinity, these $l_i$ escape
to infinity in $L$ because bigger and bigger disks
in $g_i(F_i)$ flow to $L$. 
This implies that there is no ideal point of a ray
of $\wlsl$ outside the small segment of $\pin L$
bounded by $u_0, u_1$. This contradicts
the previous proposition that such ideal points are
dense in $\pin L$.

This finishes the proof of the lemma.
\end{proof}

\begin{lemma}{}{}
The limit points 
of rays of $\wlsf$ 
vary continuously in $\pin F$
except for the
non Hausdorffness in the leaf space of $\wlsf$.
\label{encon}
\end{lemma}

\begin{proof}{}
Suppose that $p_i$ converges to $p$ in $F$, with respective rays
$r_i$ converging to the ray $r$ of $\wlsf$. 
Let $l$ be the leaf of $\wlsf$ through $p$.
Up to subsequence
assume the $r_i$ are all in the same sector of $l$ defined by $p$
and that they form a nested sequence of rays.
Then the ideal ponts 
$(r_i)_{\infty}$ form a monotone sequence
in $\pin F$. Perhaps some
ideal points are the same.
If $(r_i)_{\infty}$ does not converge to $r_{\infty}$ there
is an interval $v$ in $\pin F$, between the limit and $r_{\infty}$.
Since the ideal points are dense in $\pin F$, there is $w$ leaf
of $\wlsf$ with $w_{\infty}$ in $v$. Therefore there is $l'$ not
separated from $l$ with $r_i$ converging to $l'$ as well.
In this fashion we can go from $l$ to $l'$. 
This shows that if there is no leaf of $\wlsf$ non separated from
$l$ in that side and in the direction the rays $r_i$ go, then
the limit points vary continuously.

We analyse a bit further the non Hausdorffness.
In the setup above there are subrays of $r_i$ with points
converging to a point in $l'$ and we can restart the
analysis with $l'$ instead of $l$.
If there are finitely many leaves non separated from $l$ and
$l'$ we can assume that $l, l'$ are consecutive. Then
they have subrays which share an ideal point. 
If $m$ is the last leaf non separated from $l, l'$ 
in the direction the rays $r_i$ go to, then
there is a ray $\zeta$ of $m$ so that there are subrays
of $r_i$ with points converging to a point in $\zeta$
and $(r_i)_{\infty}$ converges to $\zeta_{\infty}$.
If there are infinitely many such leaves non separated
from $l$, then we can order them as $\{ l_j \}, j \in {\bf N}$ 
all in the
direction the rays $r_i$ go to. The ideal points of
\ $l_j$ \ form a monotone sequence in $\pin F$ which converge
to a point $u$ in $\pin F$. The arguments above show
that $(r_i)_{\infty}$ converges to $u$.
\end{proof}

Our next goal is to analyse the 
non Hausdorffness in the leaf space of $\wlsf$.
We also want to understand when can the ideal points of two
different rays of 
the same leaf of $\wlsf$ be the same.
A {\em Reeb annulus} is an annulus $A$ with a foliation so that
the boundary components are leaves and every leaf in the
interior is a topological line which spirals towards 
the two boundary components in the same direction.
In the universal cover the lifted foliation does
not have Hausdorff leaf space.
The lifted foliation to the universal cover is 
called a {\em Reeb band}.

\begin{define}{(spike region)}{}
A stable spike region in a leaf $F$ of $\fn$
is a closed $\wlsf$ saturated set ${\cal E}$
satisfying:


\begin{itemize}

\item
There are
finitely many  boundary leaves of ${\cal E}$
which are line leaves of $\wlsf$.
The ideal points of consecutive rays in the boundary of ${\cal E}$
are the same, otherwise they are distinct (like an ideal
polygon). 


\item
The region ${\cal E}$ is a bounded distance
from the ideal polygon with these vertices. The bound
is not universal in $\fn$. 

\item
There is an ideal
point  $z$ of  ${\cal E}$
so that every leaf in the interior of ${\cal E}$ has both ideal
points equal to $z$. In addition the leaves in the
interior are nested.
The finitely many leaves in the boundary
are all non separated from each other and they are limits
of the interior leaves.

\item
There is no singularity of $\wlsf$ in the interior of
${\cal E}$. If there is a singularity of $\wlsf$ 
in a boundary leaf $\tau$ of ${\cal E}$ then 
the interior of ${\cal E}$ is contained in the sector
defined by the line leaf $\tau$.
\end{itemize}

\noindent
Similarly define an unstable spike region. A spike region is either
a stable or unstable one.
\end{define}


\begin{proposition}{}{}
Let $E$ be a leaf in $\fn$ and $\upsilon$ a slice of a leaf
$\upsilon_0$ of $\wlse$.
Suppose that both ideal points of $\upsilon$ are the same.
Then $\upsilon$ is contained in 
the interior of a stable spike region $B$ of $E$.
%, so that
%$B$ has 2 distinct ideal points in $\pin E$ and $B$ is a bounded
%distance from the geodesic connecting these ideal
%points. In addition the foliation in $B$ is 
%a Reeb band.
In addition either $B$ projects to a Reeb annulus in a leaf
of $\fol$ or for any two consecutive rays in $\partial B$,
the region between them projects to a set asymptotic to
a Reeb annulus in a leaf of $\fol$.
Similarly for $\wluf$.
\label{band}
\end{proposition}

\begin{proof}{}
We do the proof for $\wlsf$.
Let $\upsilon$ be a slice as above with ideal point $x$ in 
$\pin E$. 
Let $C$ be the region bounded by $\upsilon$ in $E$ which
only limits in $x$.
First we assume 
assume that $\upsilon$ is a line leaf of some leaf of
$\wlsf$.
%$\upsilon_0$,
%since any prong of $\upsilon_0$ which enters $C$ will
%have ideal point $x$.
We will show that 
the region $C$ as it approaches $x$, projects to a set
in $M$ which limits to a Reeb annulus in  a leaf of $\fol$.
The process will be done in a series of steps.
The proof of this proposition is very long with
several intermediate results and lemmas.

Choose $z_0$ in $\upsilon$ and let $e_1, e_2$ be the rays
of $\upsilon$ defined by $z_0$.
Let $\zeta^*$ be the geodesic ray of $E$ starting at
$z_0$ and with ideal point $x$. Then $\zeta^*$ is 
contained in the $2 \delta_0$ neighborhood of $e_1$ or $e_2$,
where $\delta_0$ is the constant of lemma
\ref{boun}. It follows that 
we can choose $p_i, q_i$ in $e_1, e_2$ respectively 
with $p_i, q_i$ converging to $x$ in $E \cup \pin E$
and also $d_E(p_i, q_i) < 4 \delta_0$. 
Let $e^i_1$ be the subray of $e_1$ starting at $p_i$
and $e^i_2$ the subray of $e_2$ starting at $q_i$.
Up to subsequence  there are $p_0, q_0$ in $\mi$ and
are $g_i$ in $\pi_1(M)$ with $g_i(p_i)$, $g_i(q_i)$
converging to $p_0$, $q_0$ respectively. 
The distance condition implies 
$p_0, q_0$  are in the same leaf of $\fn$, let $F$ be this leaf.
Then $g_i(E)$ converges to $F$
and perhaps other leaves as well.

For $i$ big enough
the flowlines of $\wwp$ through $g_i(p_i), g_i(q_i)$
go through to $u_i$ and $v_i$ in $F$. Also $u_i \rightarrow p_0,
v_i \rightarrow q_0$.
If the leaf of $\wlsf$ through $p_0$ contains $q_0$ then 
for $i$ big enough the arcs 
in leaves of $\wlsf$ from $u_i$ to $v_i$
will have bounded length and
bounded diameter. The same will happen for the arcs of
of $g_i(\upsilon)$ between
$g_i(p_i)$ and $g_i(q_i)$, contradiction.
Hence $p_0, q_0$ are not in the same leaf of $\wlsf$. Let
$l$ be the leaf of $\wlsf$ through $p_0$ and $r$ be the one
through $q_0$. Let $L, R$ be leaves of $\wls$ containing $l$ and
$r$ respectively.
Since the intersection of
a leaf of $\wls$ with $F$ is connected, then $L$ and $R$ are
distinct and also are not separated from each other
in the leaf space of $\wls$.





%Fix $i$ big enough so that $g_i(p_i)$ is very close
%to $p_0$ and $g_i(q_i)$ very close to $q_0$.
%Let $A$ be the leaf of $\wls$ containing $g_i(p_i), g_i(q_i)$.
%The index $j$ will denote an arbitrary integer much bigger
%than $i$ so that $g_j(p_j)$ is much closer to $p_0$ and similarly
%for $g_j(q_j)$.
%Choose $\gamma$ a segment connecting $p_0$ in $L$ to $q_0$ 
%in $R$ and intersecting $L$, $R$ only in these points and
%also transverse to $\fn$ near $p_0, q_0$. Since $L, R$
%are non separated we can choose $\gamma$ to be transverse
%to $\fn$ except at one tangency.
%The point $g_j(p_j)$ is in $g_j \circ g^{-1}_i(A)$.
%For simplicity let $h = g_j \circ g^{-1}_i$
%(which really depends on $j$).
%We want to analyse $h^{-1}(L), h^{-1}(R)$ which are non
%separated from each other, see fig. \ref{conv}.
%Our goal is to show that $h(L) = L$ and $h(R) = R$.
%
%\blankfig{conv}{1.1}{Non separated leaves and rigidity.}
%
%
%Now fix $j$.
%We can choose $\gamma$ so that $g_j(p_j) = h(g_i(p_j))$ and
%$g_j(q_j) = h(g_i(q_j))$ are in $\gamma$.
%They are closer to $p_0, q_0$ than $g_i(p_i), g_i(q_i)$ are.
%Figure \ref{conv} is a representation of this situation
%in the universal cover $\mi$.
%In the leaf $A$, the orbits intersecting $\gamma$, that is,
%those containing $g_i(p_i)$ and $g_i(q_i)$ separate
%the orbits containing $g_i(p_j), g_i(q_j)$ $-$ reflecting
%what happens in $\alpha$.
%
%\vskip .1in
%\noindent
%{\underline {Claim 1}} $-$ The leaves $h^{-1}(L), h^{-1}(R)$
%are non separated from $L$ and $R$.
%
%First of all we show that $R$ does not intersect $h^{-1}(\gamma)$
%except perhaps at $h^{-1}(R)$ and $h^{-1}(L)$.
%Suppose not. Then since $h(A)$ is closer to $R$ then
%$A$ is, then $h^{-1}(R)$ intersects $h^{-1}(\gamma)$
%either between $h^{-1}(R \cap \gamma)$ and 
%$g_i(q_j)$
%or between $h^{-1}(L \cap \gamma)$ and 
%$g_i(p_j)$.
%If $R$ intersects $h^{-1}(\gamma)$ between $h^{-1}(L \cap \gamma)$
%and $g_i(p_j)$, then because of the ordering of orbits
%in $A$ and in $R$ it follows that $R$ has to
%intersect $\gamma$ between $g_i(p_i)$ and $L \cap \gamma$.
%But then $R, L$ intersect a common transversal, contradiction
%to these leaves being non separated in the leaf space.
%If $R$ intersects $h^{-1}(\gamma)$ between $h^{-1}(R \cap \gamma)$
%and $g_i(q_j)$ then since $h^{-1}(L)$ is non separated
%from $h^{-1}(R)$, it follows that $R$ separates 
%$h^{-1}(L)$ from $A$. This reduces to the first
%situation.
%
%Let $\xi_1$ be the ideal point of the ray of $l$ which is
%the limit of the rays associated to $e_1$ and $\xi_2$
%the ideal point of $r$ the one corresponding to $e_2$.

For simplicity assume that the leaves of $\wls$ through
$u_i$ form a nested collection with  $i$.

\vskip .1in
The first goal is to show that we can choose $l, r$ line leaves
of $\wlsf$ as above so that they also share an ideal
point.
Let $\beta_i$ be a ray in the leaf of $\wlsf$ through
$u_i$ starting at $u_i$ and containing points in the 
flowlines which go to the ray
$g_i(e^i_1)$. Similarly let $\gamma_i$ be a subray
in the same leaf starting at $v_i$ and associated with 
the ray $g_i(e^i_2)$.
Let ${\cal C}_1$ \ (resp. ${\cal C}_2$) be the collection of line leaves of 
$\wlsf$ that $\beta_i$ \ (resp. $\gamma_i$) converges to, including the
ray of $l$ \ (resp. $r$).
Let ${\cal C}$ be the collection of all line leaves of $\wlsf$ which
are non separated from $l, r$. 
Then ${\cal C}$ contains ${\cal C}_1$ and ${\cal C}_2$. 
For any element $\tau$ in ${\cal C}$, let
$B(\tau)$ be the leaf of $\wls$ containing it.
 All of the $B(\tau)$ are not separated
from each other, and they are in the set of leaves ${\cal B}$ of $\wls$
non separated from both $L, R$.
By theorem \ref{theb}, the set ${\cal B}$ has a linear order,
making it order isomorphic to either ${\bf Z}$ or a finite
set. This induces an order in ${\cal C}$ where we can choose
this so that an arbitrary element of ${\cal C}_1$ is
bigger than any element in ${\cal C}_2$.


\begin{figure}
\centeredepsfbox{gm9.eps}
\caption{
a. Non Hausdorffness in the limit,
b. Showing $\xi_1 = \xi_2$.}
\label{spre}
\end{figure}



If there are finitely many elements in ${\cal C}_1$ let 
$l'$ be the last one and let $\xi_1$ be the ideal point of
the ray of $l'$ corresponding to the direction of the
rays $\beta_i$.
Otherwise the ideal points of the leaves
in ${\cal C}_1$ form a weakly monotone sequence in $\pin F$
and let $\xi_1$ be the limit of this sequence.
Similarly define $\xi_2$ associated to $r$, see fig. \ref{spre},
a.

Fix a basepoint $x_0$ in $F$.
The first thing to prove is the following:

\begin{lemma}{}{}
$\xi_1 = \xi_2$.
\end{lemma}

\begin{proof}{}
Suppose by way of contradiction that this is not true.
Choose 2 markers very near $\xi_1$ bounding
an interval $J_1$ in $\pin F$ with $\xi_1$ in the
interior and similarly choose markers near $\xi_2$ and
interval $J_2$ so that $J_1, J_2$ are disjoint.
Let $W_1$ be the wedge of $F$ centered at the point
$x_0$ with
ideal set $J_1$ and $W_2$ the wedge of $F$ centered
also at $x_0$ with ideal set $J_2$. For i big enough
both boundaries of $W_1$ and $W_2$ flow into
$g_i(E)$. 

Suppose first that there is a last leaf $l'$ in ${\cal C}_1$.
Then $l'$ has a ray which is eventually contained in
a strictly smaller wedge $W'_1$, since 
the ideal point of $l'$ is  $\xi_1$.
%in a strictly smaller subwedge $W'_1$ of $W_1$ $-$ since its
%ideal point is $\xi_1$.
Now choose a big disk $D$ of $F$ centered in $x_0$ .
Let $N_1$ be the closure of $W_1 - D$.
Choose $D$ big
enough so that $l'$ enters $N_1$ through $\partial D$
and is then entirely in $W'_1$. 
For $i$ big enough $\beta_i$ will be close to $l'$ for 
a long distance. By lemma \ref{encon} the ideal 
points of $\beta_i$ converge to $\xi_1$ as $i$ converges
to infinity, since $l'$ is the last leaf non separated
from $l$ in that side. The ideal point is in the limit
set of the subwedge $W'_1$. If the rays $\beta_i$ keep
exiting $W_1$ then since they are trapped by $l'$ and
$\beta_{i_0}$ (for some $i_0$), it follows that  the
sequence \ $\beta_i$ \ 
%they will have to intersect a compact
%part of $\partial W_1$. Then the sequence $\{ \beta_i \}$ 
has additional limits besides the leaves in ${\cal C}_1$,
contradiction.
Therefore for big enough $i$, the $\beta_i$ enters $N_1$ through
$\partial D$ and stays in $N_1$ from then on.

We want to get the same result when ${\cal C}_1$ is infinite.
In that case let \ $\{ \nu_j, \ j \in {\bf N} \}$ 
\ be the leaves in
${\cal C}_1$ ordered with same ordering as in ${\cal C}_1$
and $\nu_1 = l$.
%$ in the same leaf as $l$.
Since the leaves $v_i$ are non separated from each other
then they cannot accumulate anywhere in $F$ as $i \rightarrow \infty$
and the leaves $\nu_j$ escape compact sets as $j$ grows.
The ideal points of $\nu_j$
are also converging to $\xi_1$. By 
density of ideal points of $\wlsf$ in $\pin F$ the
leaves 
$\nu_j$
cannot be getting closer to non trivial intervals in $\pin F$.
This implies that there is $j_0$ so that for 

$$j \geq j_0, \ \ 
\nu_j \ \ {\rm  is \ very \ close  \ to } \ \ 
\xi_1 \ \ {\rm in} \   F \cup \pin F$$

\noindent
and so contained in $W_1$.
Now an argument entirely similar as in the case ${\cal C}_1$
finite implies that for $i$ big enough then $\beta_i$ has
subrays entirely contained in $N_1$.
The same holds for $\gamma_i$ producing subrays entirely
contained in the corresponding set $N_2$ $-$ the disk 
$D$ may need to be bigger to satisfy all these conditions.

There is $a_1 > 0$ and 
$i_0$ so that for $i \geq  i_0$ then except for the initial
segment of length $a_1$ then  $\beta_i$ is entirely
contained in $N_1$ and similarly for $\gamma_i$ and $N_2$.
Choose $k_0$ big enough so that $D$ is $\epsilon$ close
to $g_k(E)$ for any $k \geq k_0$.
Then $D$ flows in $g_k(E)$ under $\wwp$
and so do $\partial W_1, \partial W_2$.
For $i$ bigger than both $i_0, k_0$
the ray $\beta_i$ flows into the ray $g_i(e^i_1)$
(notice these do not
have singularities). 
The ray $g^i(e^i_1)$ has to be in the generalized wedge
which is bounded by the image of $\partial W_1$ in $g_i(E)$.
Similarly for $\gamma_i$. 
This argument is done in lemma \ref{cont}.
These two generalized wedges
have disjoint ideal sets in $\pin g_i(E)$. Therefore
$g_i(e^i_1)$ and $g_i(e^i_2)$ do not have the same
ideal points. This is a contradiction
because $e_1, e_2$ have the same ideal point in $\pin E$. 

This proves that $\xi_1 = \xi_2$.
\end{proof}


\noindent
{\underline {Continuation of the proof of proposition \ref{band}}}


The first part of the proof was this: in $E$ zoom in
towards an ideal point $x$ of  
$\pin E$ and use covering translations $g_i$ of $\mi$
to map back these points near a point in $\mi$ which
is in a leaf $F$.
We will redo this process starting with $F$.
By taking translates of $F$ we will limit to a leaf $F^*$.
The difference is that now we have leaves
of $\wlsf$ which are non separated from each other.
These non separated leaves are much better suited
to perturbation arguments as seen below.

The lemma shows that $\xi_1 = \xi_2$ and this
implies that the ideal points
of $\beta_i, \gamma_i$ are all the same and equal to $\xi_1$.
%Next we want to show that under these conditions then
%${\cal C}_1$ has to be finite and similarly for ${\cal C}_2$.
Let $\xi = \xi_1$.
The $\beta_i, \gamma_i$ are rays in leaves of
$\wlsf$ and contained in $F$.
Let $\mu$ be the geodesic ray in $F$ starting at $x_0$ 
(the basepoint in $F$) with
ideal point $\xi$.
Since $(\beta_i)_{\infty} = (\gamma_i)_{\infty} = \xi$,
then lemma \ref{boun} implies that for $z$ in
$\mu$ far enough from $p_0$, we can choose
a point in $\beta_i$ which is less than 
$2\delta_0$ away from $z$ in $F$.
Call this point $b_i(z)$. Similarly define $c_i(z)$ in $\gamma_i$.
This is for any $i$ in ${\bf N}$. 

For each $z$ we may take a subsequence
of the $b_i(z)$ which converges in $F$ and 
the limit is denoted by $b(z)$. Similarly define $c(z)$.
By definition of ${\cal C}_1$ the point $b(z)$ has
to be in one of the leaves of ${\cal C}_1$ and
similarly for $c(z)$.
The $b(z), c(z)$ are not uniquely defined and most likely
do not vary continuously with $z$.


\begin{lemma}{}{}
There is at least one element  $\zeta$ of ${\cal C}_1$ which
has ideal point $\xi$.
Similarly for ${\cal C}_2$.
\end{lemma}

\begin{proof}{}
If there are finitely elements in ${\cal C}_1$ then the
last one satisfies this property.
Suppose then there are infinitely many elements
in ${\cal C}_1$.
As $z$ varies in $\mu$, then so does $b(z)$. If 
there are $z$ escaping in $\mu$ so that $b(z)$ is
in the same element $\zeta$ of ${\cal C}_1$ then
$\zeta$ has an appropriate ray with ideal
point $\xi$. In this case we are done.

Otherwise we can find $z_k$ in $\mu$ converging to $\xi$
so that $b(z_k)$ are in leaves $\nu_{m(k)}$ of ${\cal C}_1$
which are all distinct. We can choose $z_k$ so that
the $m(k)$ increases with $k$.
In the same way we have $c(z_k)$ in 
distinct elements of ${\cal C}_2$.
Let 

$$B_k \ = \ \ws(b(z_k)), \ \ C_k \ = \ \ws(c(z_k)),
\ \ \ {\rm both \ in} \  \ {\cal B}$$

\noindent
Recall that ${\cal B}$ is the set of leaves of $\wls$
non separated from both $L, R$.
As the length from $b(z_k)$ to $c(z_k)$ in $F$
is bounded by $4\delta_0$, then up to subsequence assume
$\pi(b(z_k)), \pi(c(z_k))$ converge in $M$.
For $n, k$  big enough there is $h_{nk}$ covering
translation of $\mi$ so that $h_{nk}(b(z_n))$ is very close
to $b(z_k)$ and $h_{nk}(c(z_n))$ is very close to 
$c(z_k)$. Suppose $n >> k$, let $h = h_{nk}$ for simplicity.
Then $B_k$ has a point $b(z_k)$  very close to 
$h(b(z_n)) \in  h(B_n)$ and similarly $c(z_k)$ in 
$C_k$ very close to $h(c(z_n)) \in h(C_n)$.
But $B_k$ is non separated from $C_k$ and
similarly for $h(B_n), h(C_n)$, so the only
way this can happen is that 

$$h(B_n) \ = \ B_k, \ \ \ \ h(C_n) \ = \ C_k$$

\noindent
This implies that $h$ sends the set of leaves non
separated from $B_i, C_i$ to itself, that is $h$ acts
on the set ${\cal C}$ and therefore acts on ${\cal B}$ as
well. 
Notice that $B_k < B_n$ in the order of ${\cal B}$ 
because $n > k$ 
and $C_k \geq C_n$ (the $C_k$ could be all the same, but
if they are not then they decrease in the order).
Since $h(B_n) = B_k$ then
$h$ acts as a decreasing translation in the ordered
set ${\cal B}$.
But since $h(C_n) = C_k$ then $h$ acts as a non
decreasing translation. 
These two facts are incompatible.

This implies that we have to have at least one element
in ${\cal C}_1$ with ideal point $\xi$.
The same happens for ${\cal C}_2$.
This finishes the proof of the lemma.
\end{proof}

%\noindent
%{\bf {Remark}} $-$ 
%As will be seen from future arguments, there is only
%one element in ${\cal C}_1$ and similarly for 
%${\cal C}_2$. These two line leaves have
%an ideal point $\xi$.
%\vskip .1in


Since the sequence $\{ \beta_i \}$ 
also converges to the leaf $\zeta$ of $\wlsf$ we can rename
the objects and assume that $l = \zeta$ and $p_0$ is
a point in $l$. This can be accomplished by
choosing different points $p_i$ in the
ray $e_1$. Similarly do the same thing in the
other direction. We state this conclusion:

\vskip .1in
\noindent
{\bf {Conclusion}} $-$  There are $p_i, q_i$ in $e_1, e_2$
respectively, escaping these rays, so that 
$d_E(p_i, q_i) < 4 \delta_0$
and there are covering translations $g_i$ so that:
$g_i(p_i)$ converges to $p_0$, 
\ $g_i(q_i)$ converges to $q_0$,
both in $F$ and in rays $l, r$ of $\wlsf$.
Also $l, r$ converge to the same ideal point
$\xi$ in $\pin F$ and $l, r$ are not separated
in the leaf space of $\wlsf$.
\vskip .1in

%The nearby leaves $l_i$ converge to $l$.
%If their limit points do not converge to $l$ then there is
%a leaf $l'$ non separated from $l$. We change this accordingly,
%assuming that $l$ is the last leaf non separated from 
%and in that side. Then the ideal points
%of $l_i$ are eventually in $\overline W_1$ and the
%piece of $W_1$ enters through $\partial D$ and are entirely
%contained in $W_1$. This means that same happens for nearby
%leaves. Hence in the near leaf to $F$,
%$\ws(q_i)$ also has a part very near $l_i \cap D$
%and then it enters $W_1$. Since it can only exit $W_1$ through
%$\partial W_1$ and both sides carry over to $F$, then
%this cannot happen. So the ideal point of $l_i$ in
%$g_i(A)$ is in the associated set to $W_1$
%and similarly for $r_i$. Hence they are different,
%contradiction.
%This proves the claim.
%
%Since $\xi_1 = \xi_2$ and $l$ is non separated from $e$,
%we obtain that $l_i$ is a collection of leaves
%in $F$ all with ideal points $\xi$ and limiting to $l, r$
%with ideal points $\xi$.




We will continue this perturbation approach. 
We want to show that the region in $F$ ``between"
$l$ and $r$  projects to a Reeb annulus of $\fol$ in $M$.
Let then $z_i$ in $l$ converging to $\xi$ and $w_i$ in
$r$ converging to $\xi$, so that $d_F(z_i,w_i)$ is always
less than $4 \delta_0$. Up to subsequence assume there
are $h_i$ covering translations
with 

$$h_i(z_i) \rightarrow z_0, \ \ \ h_i(w_i) \rightarrow w_0$$

\noindent
%Therefore $\ws(z_0), \ws(w_0)$ are not separated from
%each other. 
Notice that $h_i(L), h_i(R)$ are 
non separated from each other and $h_i(L) \rightarrow \ws(z_0), 
\ h_i(R) \rightarrow \ws(w_0)$.
The argument in the previous lemma then implies
that $h_i(L) = h_j(L), \ h_i(R) = h_j(R)$ for all $i, j$
at least equal to some $i_0$.
Discard the first $i_0$ terms
and postcompose $h_i$ with $(h_{i_0})^{-1}$ 
(that is $(h_{i_0}^{-1} \circ h_i)$)
and assume that this is the original sequence $h_i$.
This implies that
%the limits and
%they have to be very near these, it follows that 
%$h_i(L) = \ws(z_0)$ for $i$ big enough. So pull
%back by $h_i$ (here $i$ is fixed) to suppose 
$h_i(L) = L,
h_i(R) = R$ for all $i$.
So the $h_i$ are all in the intersection of
the isotropy groups of $L$ and $R$. This group is
generated by a covering translation
$h$.
Therefore there are $n_i$ with $h_i = h^{n_i}$. 
Since 
$h_i(z_i) \rightarrow z_0$ and the $\{ z_i, \ | \ i 
\in {\bf N} \}$ do not
accumulate in $\mi$ then 
$|n_i| \rightarrow \infty$.
In addition since $L, R$ are not separated from each other,
then $h$ preserves each individual line leaf, slice and
possible lift annulus of $L$.

Let $F^*$ be the leaf of $\fn$ containing $z_0, w_0$. Then
$h_i(F)$ converges to $F^*$.
Since $z_i$ is in $L$ and $h_i(L) = L$ then $h_i(z_i)$ is
in $L$ and so $z_0$ is in $L$. It follows that $L$ intersects
$F^*$.

Up to subsequence and perhaps taking the inverse of $h$,
assume that $n_i$ converges to $+\infty$.
If $h(F) = F$, then since $h(L) = L$ this produces
a closed leaf of $\Lambda^s \cap \fol$
in $\pi(F)$. Similarly 
$h(R \cap L) = R \cap L$ so produces another closed leaf in
$\pi(F)$ and together bound an annulus in $\pi(F)$ with a sequence of
leaves of $\Lambda^s \cap \pi(F)$
converging to the boundary leaves.
By Euler characteristic reasons, there can be no singularities
inside the annulus, so we conclude that the annulus in $\pi(F)$
has a Reeb foliation.

Now assume that $h(F)$ is not equal to $F$.
Let $\hp$ be the leaf space of $\fn$. This is a one dimensional
manifold, which is simply connected, but usually not
Hausdorff \cite{Ba2}. The element $h$ acts on $\hp$. An analysis of
group actions on simply connected non Hausdorff spaces
was iniatilly done by Barbot in
\cite{Ba2} and subsequently in \cite{Ro-St,Fe8}.
One possibility
is that $h$ acts freely in $\hp$. Then $h$ has an axis $\tau$
in $\hp$ which is invariant under $h$.  In general this
axis is not properly embedded, see \cite{Fe8}.
Since all the $h^{n_i}(F)$ intersect a common transversal,
then the analysis in \cite{Ba2} shows
that $F$ has to be in the axis of $h$ and 
$h^n(F)$ converges to a collection of non separated
leaves. In this case we get that $F^*$ and $h(F^*)$
are non separated from each other.


The other situation is that $h$ has fixed points in $\hp$.
In general the set of fixed points of $\hp$ is not a closed
set, but the set of points $z$ in $\hp$ so that $z$ and
$h(z)$ are not separated in $\hp$ is a closed subset
$Z$ of $\hp$ \cite{Ba2,Ro-St}.
None of the images of $F$ under $h$ can be in $Z$,
so $F$ is in a component of $\hp - Z$.
Then $h$ permutes these components. In addition
$h$ preserves an orientation in $\hp$ $-$ since
$\fol$ is transversely orientable.
Since
$h^{n_i}(F)$ all intersect a common transversal then
they have all to be in the same component $U$ of $\hp - Z$.
Let $i_0$ be the smallest positive integer
so that $h^{i_0}(U) = U$.
It follows that all $n_i$ are multiples of $i_0$.
Since $h^{n_i}(F)$ converges to $F^*$ then
the leaf $F^*$ is in the boundary of the component
$U$ and $h^{i_0}(F^*) = F^*$. 
%Therefore the same
%argument as above can be applied to show that 
%there is a Reeb annulus in $\pi(F^*)$.

The only remaining case to be analysed
is that $h$ acts freely and $h^n(F)$ converges
to $F^*$ with $h(F^*)$ non separated from $F^*$.
In this particular case we prove this is not
possible, that is:

%If $h(F)$ is not equal $F$, then  as $h^{n_i}(z_i) \rightarrow z_0$,
%it follows that
%$h^{n_i}(F)$ converges to $F^*$ which contains $z_0$ and 
%also $w_0$.
%Up to subsequence assume that
%$n_i = a \ (mod \ n_1)$ for all $i \geq 2$.
%Let $S = h^{n_2}(F)$ and $f = h^{n_1}$. Then
%
%$$h^{n_i}(F) \ = \ h^{n_i - n_2} h^{n_2}(F) \ = \ h^{m_i n_1}(S)
%\ = \ f^{m_i}(S)$$
%
%\noindent
%with $m_i$ converging to $\infty$.
%%For simplicity assume that $F = S$ and consider $f^{m_i}(F)$.
%Then $S$ is in a local axis of $f = h^{n_1}$
%and $f^n(S)$ converges to $F^*$ when $n$ converges to infinity.
%Local axis means that $f(S)$ separates $S$ from $f^2(S)$.
%Since $S$ is a translate of $F$, the structure of $\wlsf$
%gets translated into $S$.

\vskip .1in
\noindent
{\bf {Claim}} $-$ $h(F^*) = F^*$.

Suppose this is not true. 
The leaves $h(F^*), F^*$ are 
not separated in $\hp$. 
%the leaf space of $\fn$, see
%fig. \ref{jump}, a.
This implies that $\Theta(F^*)$ and $\Theta(h(F^*))$ are
disjoint subsets of $\oo$, see fig. \ref{jump}.
Therefore there are boundary leaves separating them.
But $L$ intersects both $F^*$ and $h(F^*)$
as $L$ intersects $F^*$ and is invariant under $h$.
Therefore both 
$\Theta(F^*)$ and $\Theta(h(F^*))$ intersect
the same stable leaf $\Theta(L)$.


Suppose that there is a stable boundary component of
$\Theta(F^*)$ separating it from $\Theta(h(F^*))$.
Then it has to be  a slice of $\Theta(L)$ as this
set intersects both of them. It would not be a line
leaf of $\Theta(L)$.
But as remarked before, $h$ leaves invariant all the
slices, line leaves and lift annuli of $L$ and
this contradicts $\Theta(h(F^*))$
% = \Theta(h^{n_1}(F^*))$ 
being disjoint from $\Theta(F^*)$.
This implies there is
an {\underline {unstable}} boundary component
of $\Theta(F^*)$ separating it from $\Theta(h(F^*))$,
see fig. \ref{jump}.



\begin{figure}
\centeredepsfbox{gm10.eps}
\caption{
Contradiction in the orbit space $\oo$.}
\label{jump}
\end{figure}




%In the same way $L$ intersects $f^2(F^*)$ as well and so on.
%Notice that $F^*$ does not intersect any periodic orbit in
%$L$. 
%This is because such orbits are invariant under $f$ and then
%$F^*, f(F^*)$ would intersect the same orbit of
%$\wwp$ and be separated.
%%In addition Also $f$ leaves the sectors associated to the 
%%periodic orbit invariant. This implies that line leaves
%are $f$ invariant ???? 
%In addition $F^*$ cannot intersect a lift
%annulus $A$ in $L$: if it intersects $A$ then it intersects
%all orbits of $\wwp$ in the interior of $A$.
%But these sets are invariant by $f$
%and $\Theta(F^*)$, $\Theta(f(F^*))$ are disjoint, contradiction.
%
%Now $L$ intersects $\Theta(F^*)$, $\Theta(f(F^*))$,
%$\Theta(f^2(F^*))$, etc.. Similarly $\Theta(R)$
%also intersects these same leaves. 

In the same way $\Theta(R)$ intersects both $\Theta(F^*)$ and
$\Theta(h(F^*))$.
Let $L_i = \ws(u_i)$. Recall from the
beginning of the proof of  proposition \ref{band}
that $u_i, v_i$ are  points in $F$ with $u_i$ converging
to $p_0$ in $L$ and $v_i$ converging to $q_0$ in $R$.
Then 
$\Theta(L_i)$ converges to 
$\Theta(L) \cup \Theta(R)$ (maybe other leaves as well).
So $\Theta(L_i)$ intersects $\Theta(F^*)$ and $\Theta(h(F^*))$ 
for $i$ big enough.
The intersection of $\Theta(L_i)$ with 
at least one of $\Theta(F^*)$ or $\Theta(h(F^*))$ 
cannot be connected,
see fig. \ref{jump}.
This contradicts  propostion \ref{conn}.
This contradiction implies that
$h(F^*) = F^*$ and proves the claim.

\vskip .1in
%This now has other consequences:
So far we have  proved the following:
in any case there is $i_0$ a positive integer
so that if $f = h^{i_0}$ then
$f(F^*) = F^*$. As $f(L) = L$ then
$f(F^* \cap L) = F^* \cap L$
and similarly $f(F^* \cap R) = F^* \cap R$.
This produces an annulus $B$ in 
$\pi(F^*)$ with a Reeb foliation.
The region of $F^*$ bounded by $F^* \cap R$ and
$F^* \cap L$ bounds a band $B$
which is a bounded distance from a geodesic in
$F^*$ and projects to a Reeb annulus in a leaf of $\fol$.




But to prove proposition \ref{band},
we really want these facts for $F$ and not just $F^*$.
That is, we want a region in $\pi(E)$ which spirals
towards a Reeb annulus.
This turns out to be true: $\pi(E)$ has points converging
to $\pi(F)$ and $\pi(F)$ has points converging to a Reeb
annulus in $\pi(F^*)$. Since the annulus is compact,
it turns out the second step is unnecessary.
This depends on an analysis of holonomy of the foliation
$\fol$ near the annulus in $\pi(F^*)$ as explained below.


\vskip .1in
\noindent {\bf {Claim}} $-$ The point
$\pi(p_0)$ of $\pi(F)$ is in the boundary of
a Reeb annulus of $\fol$ contained in $\pi(F)$.
This implies that $F = F^*$.

%In fact $\pi(F)$ has
%an annulus with a Reeb foliation.
%
%In $\pi(F)$ we consider the direction $\delta$
%of collapsing of ideal
%points. A closed curve which is oriented. Consider
%its holonomy of $\fol$ in the direction
%of $\delta$.

The point $z_0$ is in $F^* \cap L$. Then $\pi(z_0)$ is
in $\pi(F^* \cap L) = \alpha$ which is a closed curve
since $h^{i_0}$ leaves invariant both $F^*$ and $L$
and their intersection is connected.
Previous arguments in the proof imply that for
$i$ big enough $h_i(z_i)$ is in the same local
sheet of $\wls$ as $z_0$.
Hence the points $\pi(z_i)$ are in $W^s(\pi(z_0)) = \pi(L)$
and converge to $\pi(z_0)$. This shows that $\pi(F \cap L)$
is asymptotic to $\alpha$ in the direction corresponding
to the projection of the direction of escaping $z_i$ in
the ray of $F \cap L$.
Namely $\alpha$ has contracting holonomy (of $\fol$) in
the side the $\pi(z_i)$ are converging to and eventually
$\pi(z_i)$ is in the domain of contraction of $\alpha$.

This means that the direction of $F$ associated to the ideal
point $\xi$ is a contracting direction towards
$F^*$. The rays in the leaves
$F^* \cap L, \ F^* \cap R$ in $F^*$ are
a bounded distance from a geodesic ray in $F^*$ with
same ideal point. The contraction above implies
that the corresponding rays $F \cap L, \ F \cap R$ of $F$ 
are also a bounded distance from a ray in $F$ with ideal
point $\xi$.


Now recall the points $p_i$ in $E$. We have $g_i(p_i)$
very close to $p_0$ in the leaf $l$ of $\wlsf$.
Also $\pi(l)$ is eventually in a region contracting
towards a Reeb annulus of $\fol$.
Hence if $i$ is big enough the $g_i(p_i)$ will also
be in this region. The leaf of $\Lambda^s \cap \fol$
through $\pi(p_i)$ will
be contracted towards the Reeb annulus in that direction.
This implies that the limit of the $\pi(p_i)$ is
already in a Reeb annulus, consequently the limit
of the $g_i(p_i)$ is
already in a Reeb band. 
%
%
%
%\vskip .1in
%%\noindent
%%{\underline {Case 1}} $-$
%$\delta$ has contracting holonomy.
%
%When we are moving in $\pi(F^*)$ in the direction
%of $p$ we get arbitrarily close to $\pi(F^*)$ in that
%direction. It will eventually get into the
%domain of contraction. As the $\pi(A)$ is limiting
%on $\pi(F)$ once it gets enough to $\pi(p_0)$, it
%will also follow $\pi(F)$ until it hits this domain
%of contraction. From then on it will spiral towards $\pi(F^*)$.
%Hence $\pi(F^*) = \pi(F)$ and it already has
%the Reeb annulus. We conclude that $F^* = F$ in this
%case.
%
%\vskip .1in
%From now on assume that $F^* \not = F$.
%
%\vskip .1in
%\noindent
%{\underline {Case 2}} $-$ There are $y_i \rightarrow 0$ with
%$v(y_i) = y_i$. 
%
%Since $\pi(F)$ gets close enough to $\delta$ in this direction
%it is either $y_i$ or between $y_i$ and $y_{i+1}$ for some
%$i$. In either case moving in this direction
%will make it limit either on $z(y_i)$ or $z(y_{i+1})$ and
%not limit on $\pi(F^*)$. This case is impossible.
%
%\vskip .1in
%\noindent
%{\underline {Case 3}} $-$ $\delta$ has expanding holonomy.
%
%So $\delta^{-1}$ has contracting holonomy. Then $\pi(F)$
%limits to $\pi(F^*)$ in the opposite direction. Since $f(L) = L$
%and we are moving in the direction away from $\pi(F^*)$ we cannot have
%$\pi(F)$ limiting on $\pi(F^*)$. We conclude that this case cannot
%happen either.
%
%The final conclusion is that cases 2) and 3) do not occur, so
%case 1) has to occur, which in fact implies that $F = F^*$.
%
%
%Since the strip in $\pi(F)$ limits in $B$ and the strip
%in $\pi(A)$ limits in $\pi(F)$ 

It now follows that 
$\pi(F) = \pi(F^*)$.
That means that the second perturbation procedure (from points
in $F$ to points in $F^*$) in fact does not
produce any new leaf. 
This implies that up to covering translations then
the leaf $E$ is asymptotic to $F$ in the direction of
the ideal point $x$ in $\pin E$.
Let $V$ be the region of $E$ bounded by $\upsilon$ with
ideal point $x$. Then outside of a compact part
it projects very near a Reeb annulus in $\pi(F)$ and so 
this tail of $V$
has no singularity of the foliation $\wlse$.
By Euler characteristic reasons it follows that
the interior of 
$V$ has no singularities in the compact part of $V$ also.
In fact the arguments show that the tail of $V$ flows
into the {\underline {interior}} of a Reeb band 
in a nearby leaf $U$ of $\fn$.
Then leaves of $\wlse$ near $\upsilon$ on the outside of
$V$ will also flow to interior of the Reeb band  in $U$.
Therefore there are appropriate 2 rays on the outside 
so that they will be in the same leaf of $\widetilde \Lambda^s_U$
and hence in the same leaf of $\wls$.
It follows that in $E$ the leaf $\upsilon$ is also approximated
on the outside by a leaf which has a line leaf with 
both ideal points the same. This implies that $\upsilon$
has no singularities.





So far we proved the following:


\vskip .1in
\noindent
{\bf {Conclusion}} $-$ Let $\upsilon$ be a slice of 
$\wlse$ with 
two rays converging to the same ideal point $x$ of
$\pin E$  and $V$ is the region of $E$ bounded by $\upsilon$.
Then $\upsilon$ has no singularity of $\wlse$ and neither does $V$.
Also $\pi(V)$ is either contained in or asymptotic
to a Reeb annulus in a leaf of $\fol$ and so $E$
is asymptotic to a Reeb band in a leaf $F$ in
the direction $x$.
\vskip .1in

\begin{figure}
\centeredepsfbox{gm11.eps}
\caption{
a. $l_i$ converging to non separated
leaves $e_L, e_Z, e_Y, e_R$ of $\wlse$, b. Nested families
and identifications of ideal points.}
\label{nonsep}
\end{figure}


\noindent
{\underline {Continuation of the proof of proposition \ref{band}}}.

What we want to prove is that in $E$ itself the region
$V$ is contained in the interior of a spike region.
Notice it is not true in general that $\pi(V)$ is
contained in a Reeb annulus, only that it is
asymptotic to a Reeb annulus. For instance start
with a leaf of $\fol$ having a Reeb annulus and blow that
into an I-bundle. Then produce holonomy associated
to the core of the Reeb annulus. Then one produces
Reeb bands asymptotic to but not contained in Reeb annuli.

Since $V$ is asymptotic to the Reeb band in $F$, it turns
out that (after rearranging by covering translations) that
$E$ intersects both $L$ and $R$ leaves of $\wls$.
Their intersection produces two leaves $e_L, e_R$ of 
$\wlse$ which are not separated from each other and which
have the same ideal point $x$. There are then leaves
$l_i$ of $\wlse$ all with ideal point $x$ and which 
converge to $e_L \cup e_R$. This follows from
the fact that in $F$ the same is true and $E$ is
asymptotic to $F$ in that direction, plus the connectivity
of the intersection of $E$ with leaves of $\wls$.

Now the sequence $l_i$ can converge to other
leaves as well, all of which will be non separated
from $e_L, e_R$. 
The set of limits is an ordered set and the any other 
leaf is 
between $e_L$ and $e_R$.
By theorem \ref{theb}
there are only finitely many of them.
We refer to
fig. \ref{nonsep}, a, where for simplicity we consider there
are 4 leaves in the limit: $e_L, e_Y, e_Z, e_R$ contained
in leaves $L, Z, Y, R$ of $\wls$. These leaves of $\wls$
are non separated from each other and form an ordered set.
Let $\xi$ be the region of
$E$ which is the  union of the region bounded by all the $l_i$
plus the boundary leaves, which are non separated from $e_L, e_R$.
Clearly every leaf in the interior has ideal point $x$ and
has no singularity. We want to show that $\xi$ is
a spike region.





Any two consecutive leaves 
of $\partial \xi$ in this ordering will have rays
with same ideal point and leaves $l_i$ converging to them.
This situation is important on its own and is analysed in
the following proposition:


\begin{proposition}{}{}
Suppose $v_1, v_2$ are non separated leaves in $\wlsg$ for some $G$ 
leaf of $\fn$.
Suppose there are no leaves non separated from $v_1, v_2$ in
between them. Then
the corresponding rays 
of $v_1, v_2$ 
have the same ideal point in $\pin G$. In addition
they are a bounded distance from a geodesic ray of $G$ with same ideal point.
In $M$ this region either projects to or is asymptotic to a Reeb
annulus.
\label{nh}
\end{proposition}

\begin{proof}{}
We do the essentially the
same proof as in the case of leaves of $\wlsg$
with same ideal points, except that we
go in the direction of the non Hausdorfness. 
Because there are no non separated leaves in between
$v_1, v_2$, then the corresponding rays have the same ideal
point.
Choose $w_i, y_i$ in these rays of $v_1, v_2$ and escaping
towards the ideal point and so that $d_G(w_i,y_i)$ is
less than $4 \delta_0$.
We do the limit analysis 
using $f_i(w_i), f_i(y_i)$ converging in $\mi$. Because 
$v_1, v_2$ are non separated it follows that $f_i(w_i), f_j(w_j)$
are in the same stable leaf (of $\wls$) for $i, j$ big enough.
Hence we can readjust so that they are all in the same
stable leaf and similarly for $f_i(y_i)$. The same arguments
as before show that that region of $G$ between $v_1, v_2$ projects
in $M$ to set in a leaf of $\fol$
which is either contained in or asymptotic to
a Reeb annulus. The results follow.
In general nothing can be said about the other direction
in the leaves
$v_1, v_2$: in particular it does not follow at all 
that the other rays of $v_1, v_2$ have to have
the same ideal point.
\end{proof}




%That is $\pi(F)$ has a Reeb annulus which the strip
%in $\pi(E)$ limits to.
%This implies that $h(F) = F$ 
%(because $h^{n_1}(F) = F$ and the argument above with
%$\Theta(L) \cap \Theta(F^*)$ not empty).
%So now $g_i(A)$ limits to $F$. The limiting is obtained in
%the converging direction. Since $h(F) = F$ then $F$ does
%not intersect
%the annuli or periodic orbits.
%But
%
%$$\ws(\widetilde \alpha) \cap F \not = \emptyset,
%\ws(\widetilde \alpha') \cap F \not = \emptyset,
%\wu(\widetilde \beta) \cap F \not = \emptyset$$
%
%\noindent
%
%We analyse the situation in the Reeb annulus $A$ in $\pi(F)$ which
%is associated to the limiting behavior of $\pi(V)$ ($V$ is
%the region in $E$ bounded by a leaf $\upsilon$ with same
%ideal points in $\pin E$).
%The holonomy of $\Lambda^s$ in the positive flow 
%direction (the flow $\Phi$)
%is expanding $-$ since transverse to $\Lambda^s$ we
%have unstable leaves. One boundary component of $A$
%is in $\pi(F \cap L)$, hence is isotopic to a closed
%orbit of $\Phi$ in $\pi(L)$.
%This closed curve inherits an orientation from the positive
%direction of that orbit. This is called the {\em flow direction}.
%Similarly for the other component of $\partial A$ which 
%is in $\pi(F \cap R)$.
%Because of the above fact, the spiralling of the interior
%leaves of the Reeb annulus $A$ limits to the boundary
%leaves in the direction opposite to the flow direction,
%see fig. \ref{direc} a.
%Keeping track of directions of spiraling and holonomy
%contraction will be crucial for all the analysis here.
%
%
%\blankfig{direc}{1.1}{a. Direction of spiraling in the
%leaves of the Reeb annlus $A$ in $\pi(F)$ is opposite
%to the flow direction and also is the same
%as the direction of contracting holonomy transverse
%to $\fol$, b. The direction of holonomy in $\pi(G)$ is
%the same as the flow direction and opposite
%to the spiraling in the Reeb annulus in $\pi(G)$, c. The direction
%of holonomy contraction in $\pi(H)$ is opposite
%to the flow direction and the same as the direction
%of spiraling in the Reeb annulus in $\pi(H)$.}
%
%
%On the other hand the region in $\pi(E)$ associated to $\pi(V)$
%limits to the Reeb annulus $A$ in $\pi(F)$. This limits in
%the direction associated to the identified ideal points
%of the leaf $\pi(v)$ $-$ this direction also corresponds
%to the opposite of the flow direction, see fig. \ref{direc}, a.
%Hence $\fol$ has contracting holonomy along $A$ in the side
%$\pi(V)$ is limiting on and in the direction opposite to
%the flow direction.
%
%
%We will now continue this analysis along the leaves of $\wlse$
%non separated from $e_L$ and in the direction from $e_L$ towards
%$e_R$ passing through $e_Z, e_Y, etc..$.
%Recall that $h$ is the generator of the isotropy group
%of $L$ and $R$.
%What we just did was to analyse the situation in $e_L$ as
%one converges to the ideal point $x^*$. 
%Next look at the other ray of $e_L$ converging to the ideal
%point $x_1$ in $\pin E$.
%This ray of $e_L$ is non separated from a ray of $e_Z$ 
%which also has ideal point $x_1$ and both are a bounded
%distance from a geodesic ray in $E$. By proposition \ref{nh},
%the region between these rays projects in $M$ to a
%set spiraling to a Reeb annulus $B$ in a leaf of
%$\fol$. Lifting to $\mi$ this produces a leaf $G$
%towards which $E$ is asymptotic to in that direction
%and so that $B$ is contained in $\pi(G)$.
%The proof of proposition \ref{nh} also shows that
%the ray of $\pi(e_L)$ in question spirals towards
%a closed curve $\sigma_1$ of $\pi(L)$ . 
%Also $\sigma_0$ is isotopic to $\sigma_1$ in $\pi(L)$,
%which then implies that $G$ is invariant by $h$ as well.
%
%Now the spiraling of this ray of $\pi(e_L)$ towards
%$\sigma_1$ is in the direction opposite of the
%spiraling of (the other ray of) $\pi(e_L)$ towards
%$\sigma_0$ in $\pi(F)$. This is because 
%$\sigma_0, \sigma_1$ are both closed curves
%in the stable leaf 
%
%The spiraling direction in $\pi(F)$ lifts to the 
%direction contrary to the flow direction, see fig. \ref{direc}, c.
%The annulus in $\ws(\alpha)$ from $\pi(F)$ to $\alpha$
%intersect $\pi(F)$ and all orbits flow asymptotically
%to $\alpha$. Therefore if there is no other closed leaf
%of $\fol$ in this annulus the leaves spiral towards $\pi(F)$
%in the positive flow direction, opposite to the spiralling
%direction, contradiction.
%
%
%
%
%This means that in $A$ we are getting closer to $F$ as one goes
%in the opposite direction of the spiralling. This implies that
%the picture in $A$ is as desired in the proposition.
%If there is another compact leaf then $\pi(A)$
%has this region between two flow homeomorphic annuli,
%see fig. \ref{direcs}, a.
%Therefore it also has the same picture as in $F$.
%If $\pi(A)$ is on the negative side of $F$,
%then it intersects the $\wu(\widetilde \beta)$.
%Notice $\pi(\beta)$ is freely homotopic to
%the inverse of $\pi(\alpha)$. Hence now the 
%spiralling direction of $\pi(F)$ agrees with the flow direction
%of $\beta$.
%Now $\pi(A)$ spirals towards $\pi(F)$ in the opposite direction
%of the spiralling direction. This produces the same picture as above.
%
%\blankfig{direcs}{1.1}{a. Spiralling when directions go, b. The other case.}
%
%
%In either case we produce the same picture as desired. This finishes
%the proof of the proposition.
%\end{proof}
%
%Just to clarify what we achieved: if $A$ has 2 rays with identified ideal points,
%then it is non singular and they are part of a non Hausdorff strip which 
%is a bounded distance from a geodesic. In $M$ either $\pi(C)$
%is a Reeb annulus, or spirals towards a Reeb annulus in both directions.
%This is the only thing that can happen when ideal points agree.
%
%What about non Hausdorff behavior? A lot can be said about that
%too $-$ one has the following result:
%

Given this last proposition then for any two consecutive
rays in $\partial {\cal E}$ it follows that they are a bounded
distance from a geodesic ray in $E$. 
All that is needed to show that ${\cal E}$ is
a spike region is to prove that the ideal
points of the rays in the boundary are distinct except
for consecutive rays.

Suppose there are other identifications of ideal points of
leaves in the boundary of ${\cal E}$. Then there is at least one
line leaf $\tau$ in the boundary of ${\cal E}$ so that
$\tau$ has identified ideal points.
Our analysis so far shows that $\tau$ is in the interior
of another region similar to the one constructed above
so that all leaves have just one common ideal point.
Since the $l_i$ limit on $\tau$, then the ideal
point of $\tau$ has to be $x$. In addition the leaves in
this new region have to be nested. But if the $l_i$ together
with $\tau$ are a nested family of leaves of $\wlsf$, 
then the $\tau$ is outside the $l_i$ hence the region
in $E$ bounded by $\tau$ enclosed the whole region
${\cal E}$, see fig. \ref{nonsep}, b.
There is at least one other leaf $\tau'$ in
$\partial {\cal E}$. The same arguments we used for $\tau$
can be applied to $\tau'$. But it is impossible that
the $l_i$ are also nested with the $\tau'$, see fig. 
\ref{nonsep}, b.


This shows that the ideal points of ${\cal E}$ are
distinct except as mandated by consecutive rays. In addition
any line leaf in the boundary of ${\cal E}$ has distinct
ideal points and rays which are a bounded distance from
geodesic rays. It follows that the whole leaf is
a bounded distance from a geodesic in $E$.
This shows that 
${\cal E}$ is a spike region.
This finishes the proof of proposition \ref{band}.
\end{proof}

Finally in the case $\wls$ has Hausdorff leaf space
one can say much, much more about metric properties
of leaves of $\wlsf$:

\begin{proposition}{}{}
Suppose that $\Phi$ is an almost pseudo-Anosov flow transverse
to a foliation $\fol$ with hyperbolic leaves.
Suppose that $\wls$ has Hausdorff leaf space.
%and that $\fol$ is not
%topologically conjugate to the stable foliation of a suspension
%Anosov flow . 
Then there is $k_0 > 0$
so that for any $F$ leaf of $\wls$, then the slice leaves 
of $\wlsf$ are uniform $k_0$-quasigeodesics.
\end{proposition}

\begin{proof}{}
If there is a leaf $F$ of $\fn$ and a slice leaf of $\wlsf$
with only one ideal point, then the proof of proposition
\ref{band} shows that there are leaves of $\wls$ non
separated from each other. This is impossible.

Suppose now that for any integer $i$, there are 
$x_i$ in $\mi$, $x_i$ in leaves $F_i$ of $\fn$ with
$x_i$ in line leaves $l_i$ of $\widetilde {\Lambda}^s_{F_i}$
with distance
from $x_i$ to $l^*_i$ in $F_i$ going to infinity.
Here $l^*_i$ is the geodesic in $F_i$ with same ideal
points as $l_i$.
Up to covering translations assume $x_i$ converges
to $x$.
Also assume all $x_i$ are in the same sector 
of $\wls$ defined by $x$. Since $l_i$
converges to $l$, the arguments
in lemmas \ref{encon} and \ref{cont} would show that
the ideal points of $l$ are the same. 
This was just disproved above.

Given that, the line leaves are within some global distance
$a_0$ 
of the respective
geodesics in their leaves.
It is well known that these facts imply that
the slice leaves of $\wlsf$ are uniform quasigeodesics.
For a proof of this well known fact see 
for example \cite{Fe-Mo}.
\end{proof}


\section{Continuous extension of leaves}

The purpose of this section is to prove the main theorem: \
the continuous extension property for leaves of foliations
which are almost transverse to quasigeodesic singular pseudo-Anosov
flows in atoroidal $3$-manifolds. As seen
before this implies that $M$ has negatively curved
fundamental group.


Suppose first that $\Phi$ is an almost pseudo-Anosov flow
which is transverse to a foliation $\fol$ 
with hyperbolic leaves
in a
general closed $3$-manifold $M$.
Given a leaf $F$ of $\fn$
we introduce geodesic ``laminations" in $F$  
coming from $\wlsf, \wluf$.
We only work with the stable foliation, similar results hold for the
unstable foliation.
Assume that a leaf $l$ of $\wlsf$ is not singular. If both ideal
points are the same let $l^*$ be empty. Otherwise let $l^*$ be the geodesic
with same ideal points as $l$. If $l$ is singular, then no line leaves
of $l$ 
have the same ideal point by proposition \ref{band}. For each line
leaf $e$ 
of $l$
let $e^*$ be the corresponding geodesic and $l^*$ their
union. Let now $\tau^s_F$ be the union of these geodesics
of $F$.
Leaves of $\wlsf$ do not have transverse intersections and
therefore the same happens for leaves of $\tau^s_F$.

Suppose that $\wlsf$ has non separated leaves 
$l, v$ which are not in the boundary of a spike region.
Then there are $l_i$ converging to $l \cup v$ (and maybe
other leaves as well),
but $l^*_i$ 
does not converge to $l^*$ or $v^*$. Notice none of the limit leaves
can have identified ideal points, because then they
would be in the interior of a spike region (proposition 
\ref{band}) and have a neighborhood which is product
foliated.
Let $\otsf$ be the closure
of $\tau^s_F$. Then $\otsf$ is a geodesic lamination in $F$.
Similarly define $\tau^u_F$, $\overline \tau^u_F$.
In a complementary region $U$ 
of $\otsf$ associated to non Hausdorffness, there
is one boundary component which is added (a leaf of $\otsf - \tau^s_F$)
and which is the limit of the $l^*_i$ as above.
All of the other boundary
leaves of the region
are associated to the non separated leaves
of $\wlsf$ and are in $\tau^s_F$.


\begin{lemma}{}{}
The new leaves in $\otsf$ (that is those
in $\otsf$ - $\tau^s_F$) come from non Hausdorff
pairs $(l,v)$ 
of $\wlsf$
as in the description above.
\end{lemma}

\begin{proof}{}
Let $e_i$ in $\tau^s_F$ converging to $e$ not in $\tau^s_F$.
Then choose $l_i$ line leaves in $\wlsf$ with
$e_i = l^*_i$. Given $u$ a point in $e$, there is
$u_i$ in $l^*_i$ very close to $u$. Then there are
$p_i$ in $l_i$ which are $2 \delta_0$ close to $u_i$. 
Up to subsequence
assume that $p_i$ converges to $p_0$ and
let $l$ be the line leaf of $\wlsf$ that the sequence
$l_i$ converges to. 
Since the $l^*_i$ converges to $e$ which is not in
$\otsf$ and $l^*$ is in $\otsf$, it follows that 
$l^*_i$ does not converge to $l^*$.
By lemma  \ref{encon} this is associated to 
a non Hausdorff situation:
$l_i$ converging to $l$ and other leaves as well and $l^*$ is
the added leaf associated to this non Hausdorfness. This finishes
the proof of the lemma.
\end{proof}

\begin{lemma}{}
The complementary regions of $\otsf$ are 
ideal polygons associated to singular leaves and non
Hausdorff behavior of $\wls$.
If $M$ is atoroidal 
then these regions
are finite sided ideal polygons.
\end{lemma}

\begin{proof}{}
Let $x$ be in a complementary region $U$ of $\otsf$.
Let $e$ be a leaf in the boundary $\partial U$.
Let $I$ be the interval of $\pin F - \partial e$ containing
other ideal points of $U$.
Suppose first that $e$ is an actual leaf of $\tau^s_F$,
which comes from a line leaf $l$ of $\wlsf$.
It may be that $l$ is contained in a singular leaf $z$ of
$\wlsf$ which
is singular on the $x$ side.  This means that $z$ has ideal
points in $I$.
In that case $x$ is in the complementary region obtained
by splitting $z$. This region must be $U$.
Otherwise $l$ is not singular on the side containing
$x$ and we may assume there
are $l_i$ leaves of $\wlsf$
with ideal points in the closure of $I$ in $\pin F$,
with $l_i$ converging to $l$. If
the ideal points of $l_i$ converge to that of $l$ then eventually
$l^*_i$ separates $x$ from $e$ and $x$ is
not in the complementary region $U$ $-$ impossible. 
Hence the ideal points of $l_i$ do not
converge to $\partial e$ and there is non Hausdorfness and
a complementary region in that side of $l$. Then $x$ needs to be
in this complementary region (which is $U$) and $e$ is a boundary
leaf of $U$ which comes from a line leaf of $\tau^s_F$. 

Suppose now that $e$ is an added leaf.
There are $l_i$ 
leaves of $\wlsf$ with $e_i = l^*_i$ converging to $e$ on the side
opposite to $x$, otherwise $x$ is not in $U$.
Then $l_i$ converges to more than one leaf of $\wlsf$ 
producing non Hausdorff behavior
and a complementary region with $e$ in its boundary.
The $x$ is in the region
associated to this non Hausdorff behavior, so
the complementary region must be $U$. 

If there is a complementary region of $\otsf$ with infinitely
many sides then it is associated to non Hausdorff behavior
and so there are leaves $l_i$ of $\wlsf$ converging to infinitely
many distinct leaves of $\wlsf$.
Then there is $L$ leaf of $\wls$ which is non separated from
infinitely many other leaves. Theorem \ref{theb} implies that
there is a ${\bf Z} \oplus {\bf Z}$ subgroup
of $\pi_1(M)$, contradiction.
% and hence $\pi_1(M)$ is not negatively 
%curved, contradiction. 
This finishes the proof.
%%converging to $l$ on the other side. This produces
%a complementary region and $x$ needs to be in these.
\end{proof}

We now turn to the continuous extension property.
%A preliminary analysis of the continuous extension property
%was done in \cite{Fe6}
%in the case that $\Lambda^s, \Lambda^u$ where quasi-isometric
%singular foliations, $\fol$ is a finite depth foliation,
%and $\Phi$ is a pseudo-Anosov {\underline {transverse}} to $\fol$. 
%Under these conditions it was shown in \cite{Fe6}
%that leaves of $\wlsf, \wluf$ are 
%uniform quasigeodesics in their respective leaves $F$.
%Here we are analysing a much more general situation: 
%in particular there are examples where
%$\wls, \wlu$ have non Hausdorff leaf space \cite{Mo5,Fe6}
%and so 
%$\wlsf, \wluf$  can have non Hausdorff leaf space, immediately implying
%that their leaves cannot be uniform quasigeodesics.
%In addition the results here apply to general foliations, for instance
%to foliations with dense leaves, foliations
%with one sided branching, etc.. The main theorem is:


\begin{theorem}{(Main theorem)}{}
Let $\fol$ be a foliation in $M^3$ closed, atoroidal.
%with
%$\pi_1(M)$ negatively curved.
Suppose 
that $\fol$ is almost transverse to a 
quasigeodesic, singular pseudo-Anosov flow $\Phi_1$
and transverse to an associated almost pseudo-Anosov flow $\Phi$.
Singular means $\Phi_1$ is not a topological Anosov flow.
Then for any leaf $F$ of $\fn$, the inclusion map
$\Psi: F \rightarrow \mi$ extends to a continuous map

$$\Psi: \ F \cup \pin F \ \rightarrow \ \mi \cup \si$$

\noindent 
The map
$\Psi$ restricted to $\pin F$, gives a continuous parametrization
of the limit set of $F$, which is then locally connected.
\label{exten}
\end{theorem}

\begin{proof}{}
The hypothesis imply that $\pi_1(M)$ is negatively curved.
Difficulties in the proof  of this result are that
$\wls, \wlu$ may have non Hausdorff leaf space \cite{Mo5,Fe6}
and so 
$\wlsf, \wluf$  can have non Hausdorff leaf space. This
implies
that the leaves of $\wlsf, \wluf$ 
cannot be uniform quasigeodesics in $F$.
In addition the leaves of $\wls, \wlu$ are not quasi-isometrically
embedded in $\mi$.
The proof is done in two steps: first we define an extension and then
we show that it is continuous.

The proof will fundamentally use the fact that $\Phi_1$ is
a quasigeodesic pseudo-Anosov flow.
It was proved in \cite{Fe-Mo} that this implies
that $\Phi$ is a quasigeodesic flow as well.
From now on we use the stable/unstable foliations
$\wls, \wlu$ of $\wwp$.
First we need to review some facts about quasigeodesic 
almost pseudo-Anosov flows. 
If $\gamma$ is an orbit of $\wwp$ then it is
a quasigeodesic and hence has unique distinct
ideal points $\gamma_-$ and $\gamma_+$ 
in $\si$ corresponding to
the positive and negative flow directions \cite{Th1,Gr,Gh-Ha,CDP}.
Hence given $x$ in $\mi$ define

$$\eta_+(x) \ = \ \gamma_+, \ \ \ \eta_-(x) = \ \gamma_-,
\ \ \ \eta_+(x) \ \not = \ \eta_-(x),$$



\noindent
where $\gamma$ is the $\wwp$ flowline through $x$.
If $L$ is a leaf of $\wls$ or $\wlu$ and $a$ is a limit point
of $L$ in $\si$, then there is an orbit $\gamma$ of $\wwp$ 
contained in $L$ with either $\gamma_- = a$ or $\gamma_+ = a$,
that is, any limit point of $L$ is a limit point of one
of its flow lines \cite{Fe2,Fe6}.
Also any such $L$ in $\wls$ is Gromov negatively curved
%in the large 
\cite{Gr,Gh-Ha,Fe2,Fe6} and
has an intrinsic ideal boundary $\partial L$ 
consisting of a single forward ideal point and distinct negative
ideal points for each flow line
\cite{Fe2,Fe6}. The set $L \cup \pin L$ is a natural
compactification of $L$ in the Gromov sense.
For instance if $L$ is a non singular leaf,
then $L \cup \pin L$ is a closed disk. In this
case the foliation by flow lines in $L$ is equivalent
to the foliation in $\hh$ by geodesics sharing a fixed
point in $\su$.


A very important fact for us is that  the inclusion

$$\kappa: L \rightarrow \mi \ \ \ \ 
{\rm extends \  to \ a \ continuous \
map} \ \ \ \  \kappa: L \cup \partial L \rightarrow \mi \cup \si,$$

\noindent
\cite{Fe2,Fe6}.
This all follows from the fact that $\Phi$ is quasigeodesic.
This works for any $L$ in $\wls$ or $\wlu$.
If $L$ is in $\wls$
there is a unique distinguished ideal point in $\si$ denoted
by $L_+$
which is the forward limit point in $\si$ of any flow line
in $L \subset \mi$.
If in addition $\Lambda^s$ is a quasi-isometric singular
foliation, then the extension $\kappa$ is always a homeomorphism
into its image, but this is not true if $\Lambda^s$ is
not quasi-isometric.
Similarly for $L$ in $\wlu$.

Throughout the proof we fix a unique identification
of $\mi \cup \si$ with the closed unit ball in
${\bf R}^3$.
The Euclidean metric in this ball induces the visual distance
in $\mi \cup \si$.
Then $diam(B)$ denotes the diameter in this distance for
any subset $B$ of $\mi \cup \si$.
A notation used throughout here is the following: 
if $A$ is a subset of a leaf $F$ of $\fn$, then $\overline A$
is its closure in $F \cup \pin F$. 
%If $A$ is a subset
%of $\mi$, then $\overline A$ is its closure in $\mi \cup \si$.
%Whenever needed we will make explicit where the closure
%is being taken.



We now produce an extension $\Psi: \pin F \rightarrow \si$.

\vskip .1in
\noindent
{\underline {Case 1}} $-$ Suppose that $v$ in $\pin F$
 is not an ideal point
of a ray in $\wlsf$ or in $\wluf$.

Since $\pi_1(M)$ is negatively curved, then
complementary regions of $\otsf$ are finite sided ideal
polygons.
Hence there are $e_i$ in $\otsf$ 
so that $\{ e_i \cup \partial e_i \}, \ i \in {\bf N}$ define
a neighborhood basis of $v$ (in $F \cup \pin F$) 
and $\{ e_i \}$ forms a nested
sequence.
Here $\partial e_i$ are the ideal points of $e_i$ in $\pin F$.
We say that the $\{ e_i \}$ define a neighborhood basis at $v$.
Assume that no two $e_i$ share an ideal point $-$
possible because of hypothesis.
If $e_i$ is in $\otsf - \tau^s_F$ then it is the limit of leaves
in $\tau^s_F$ and by adjusting the sequence above we can
assume that $e_i$ is always in $\tau^s_F$.
Let $l_i$ in $\wlsf$ with $l^*_i = e_i$
and $L_i$ leaves of $\wls$ with $l_i \subset L_i$.
%By changing $l_i$ slightly we may assume that the
%$L_i$ are not singular leaves and similarly for the $B_i$.

Similarly there are $c_i$ in $\otuf$ defining a neighborhood
basis of $v$.
Up to subsequence we may assume that $e_1, c_1, e_2, c_2$, etc..
are nested and none of them have any common ideal points
(in $F \cup \pin F$)
and $c_i$ is in $\tau^u_F$.
Let $b_i$ in $\wluf$
with $b^*_i = c_i$ and $B_i$ leaves of $\wlu$ with $b_i \subset B_i$.
%We may assume that $l_i$ are not singular. Also $b_i$ in $\wluf$
%with $b^*_i = c_i$.
%and $B_i$ in
%$\wlu$ with $b_i \subset B_i$.





At this point we need the following result:

\begin{lemma}{}{}
Let $L$ leaf of $\wls$, $B$ leaf of $\wlu$ and $F$ leaf of $\fn$
so that $F$ intersects both $L$ and $B$: $l = L \cap F,
\ b = F \cap B$. Suppose that $b$ and $l$ are disjoint
in $F$. Then $L$ does not intersects $B$ in $\mi$.
\end{lemma}

\begin{proof}{}
Suppose not. Recall that $\Theta(L), \Theta(B)$ are
finite pronged, non compact
trees and they intersect in a compact subtree.
The union is also a finite pronged tree.
In addition $\Theta(L \cap B)$ is connected.
The sets $\Theta(l), \Theta(b)$ are disjoint in this union.
Let $x$ be a boundary point of $\Theta(l)$ which is
either in $\Theta(L \cap B)$ or separates $\Theta(L \cap B)$
from $\Theta(l)$ in this union, see fig. \ref{inter}, a.


\begin{figure}
\centeredepsfbox{gm12.eps}
\caption{
a. Obstruction to intersections of leaves,
b. The case of $F$ escaping up.}
\label{inter}
\end{figure}



Let $\gamma = x \times \rrrr$, an orbit of $\wwp$.
The first possibility is that $F$ escapes up
as $\Theta(F)$ approaches $x$.
Then $\gamma$ is a repelling orbit with respect to the
$\Theta(l)$ side, see fig. \ref{inter}, b
and $\gamma$ is in the boundary of a lift annulus
$A$.
This means that $\Theta(l)$ is a generalized unstable prong
from the point of view of $x$.
By proposition \ref{bounda} there is a stable slice
$r$ of $\oos(x)$ with $r$ contained in $\partial \Theta(F)$
and $F$ escapes up as $\Theta(F)$ approaches $r$, see
fig. \ref{inter}, a.
 The two sides of $r$ are the closest
generalized prongs to $\Theta(l)$ on either side of $\Theta(l)$.
This implies that $r$ separates $\Theta(b)$ from $\Theta(F)$
see fig. \ref{inter}, a.
Then $\Theta(b)$ cannot be contained in $\Theta(F)$, contradiction.

The second option is that $F$ escapes down as $\Theta(F)$ 
approaches $x$ along $\Theta(l)$.
Here there is a slice $r$ of $\oou(x)$ with $r$ contained
in $\partial \Theta(F)$ and the closest to $\Theta(l)$ 
on both sides of $\Theta(l)$. 
Either $\Theta(b) \subset r$ or $r$ 
separates $\Theta(b)$ from $\Theta(F)$. In any case
$\Theta(b)$ does not intersect $\Theta(F)$, again
a contradiction.
This finishes the proof of the lemma.
\end{proof}


\vskip .1in
\noindent
{\underline {Claim}} $-$ Both $L_i$ and $B_i$ escape in $\mi$.

Notice $e_i \cap c_j = \emptyset$ for any $i, j$.
If $l_i \cap b_j$ is non empty with $j > i$, then the nesting
property above implies that $b_{i+1}, b_{i+2}, ..., b_j$
all have to intersect. Since there is a global upper
bound on the number of prongs of leaves of $\wls, \wlu$, this
can happen for only finitely many times. Up to taking a 
further subsequence we may assume that all the $l_i, b_j$
are disjoint. 


The lemma shows that 
$L_i \cap B_j$ is empty for any $i, j$, and they
form nested sequences of leaves in $\mi$.
Suppose that the sequence $\{ L_i \}$ does not escape
compact sets. Then there is $L$ in $\wls$ which is a limit
of $L_i$ (and possibly other leaves as well).
Let $\alpha$ be an orbit in $L$ which is not in
a lift annulus. Then $\wu(\alpha)$ is transverse
to $L$ in $\alpha$ and hence intersects $L_i$ for
$i$ big enough. Since the $L_i, B_j$ are nested
this would force $\wu(\alpha)$ to intersect
$B_j$ for $j$ big enough, contradiction.
It follows that both $L_i$ and $B_j$ escape compact sets
as $i, j \rightarrow \infty$.

Let $r$ be a geodesic ray in $F$ with ideal point $v$.
For each $i$, there is a subray of $r$ contained in
the component of $F - l_i$ which is in a small neighborhood
of $v$. Hence $\Psi(r)$ has a subray which is contained 
in the corresponding component $V_i$ of $\mi - L_i$.
These components $V_i$ form a nested sequence. The
ray $\Psi(r)$ can only limit in the limit set of $V_i$.
We need the following lemma which will be 
a key tool 
throughout the proof.

\begin{lemma}{(basic lemma)}{}
Let $\{ Z_i \}$ be a sequence of leaves or line leaves
or slices or any flow saturated sets in 
leaves of either in $\wls$ or
$\wlu$ (not all leaves $Z_i$
need to be in the same singular foliation).
If the sets $Z_i$ escape compact sets in $\mi$, then
up to taking a subsequence $\overline Z_i$ converges 
to a point in $\si$.
\label{narrow}
\end{lemma}

\begin{proof}{}
Let $Y_i$ be the leaf of $\wls$ or $\wlu$ which contains $Z_i$.
Up to subsequence assume $Y_i \in \wls$.
The statement is equivalent to $diam(Z_i)$ converges to $0$.
Otherwise up to subsequence we can assume $diam(Z_i) > a_0$
for some $a_0$ and all $i$ and hence no subsequence
can converge to a single point in $\si$.
%Suppose then that $diam(Z_i) > a_0$ for all 
%$i$ and $a_0 > 0$.
%Up to subsequence assume $Z_i$ are (say) all stable and
%that $(Z_i)_+$ converges to $p$ a point in $\si$.
%For big enough $i$,
Then there is $p_i$ in $Z_i$ with visual distance from
$p_i$ to $(Y_i)_+$ is bigger than $a_0/2$.
Notice that $(Y_i)_+$ is a point in $\overline Z_i$.
Let $\gamma_i$ the orbit of $\wwp$ through $p_i$.
If $(\gamma_i)_-$ is very close to $(\gamma_i)_+ = (Y_i)_+$
then the geodesic with these ideal points has
very small visual diameter. Since $\gamma_i$ is a global
bounded distance from this geodesic \cite{Gr,Gh-Ha,CDP},
the same
is true for $\gamma_i$ contradiction to the choice of $p_i$.
Hence the geodesic above intersects a fixed
compact set in $\mi$ and so does $\gamma_i$. This
contradicts the fact that $Z_i$ escape compact
sets in $\mi$ and finishes the proof.
\end{proof}




%\vskip .1in
%\noindent
%{\underline {Claim}} $-$ 
We claim that the limit sets of $V_i$ above shrink to a single
point in $\si$.
The limit sets form a weakly monotone decreasing sequence, because
the $L_i$ are nested and so are the $V_i$.
If the limit set does not have diameter going to zero,
then there are points in the limit set of $L_i$ which
are at least $ 2 \delta_1$ apart for some fixed
$\delta_1 > 0$.
By the previous lemma the $L_i$ cannot escape 
compact sets in $\mi$, contradiction.
%So one of them is at least $\delta_0$ from $L_+$.
%There is a flow line $\gamma_i$ of $L$ with these ideal points
%and since it is a uniform quasigeodesic, it is a bounded
%distance from a geodesic of $\mi$ connecting these ideal points.
%Then $\gamma_i$ has to intersect a fixed compact set in $\mi$.
%This contradicts the fact that $L_i$ escapes compact sets
%in $\mi$.
Since the limit sets of $V_i$ shrinks to a point in $\si$, let
$\Psi(v)$ be this point. Clearly $\Psi(r)$ limits
to this point and so does $\Psi(r')$ for any other geodesic
ray $r'$ in $F$ with ideal point $v$. 



\vskip .15in
\noindent
{\underline {Case 2}} $-$ Suppose that $v$ is an ideal point
of a leaf of $\wlsf$ or $\wluf$.

Let $l$ be a ray in say $\wlsf$ which limits on $v$
and $r$ a geodesic ray on $F$ with ideal point $v$.
Then $l$ is contained in $L$ leaf of $\wls$. 
Either $\Theta(l)$ escapes in $\Theta(L)$ or limits
to a point $x$ in $\Theta(L)$.

Consider the first case. Then in the intrinsic geometry of
$L$, the ray $l$ converges to the positive ideal point
of $L$, hence in $\mi \cup \si$, the image $\Psi(l)$ converges
to $L_+$.
In the other option let $\beta = x \times \rrrr$,
an orbit of $\wwp$.
As $l$ escapes in $F$ then in $L$ it either escapes up or
down. If it escapes down then it converges to the
negative ideal point of $\beta$ in $L \cup \pin L$
and hence $\Psi(l)$ converges to $\beta_-$. Otherwise
$l$ escapes up in $L$ as $\Theta(l)$ approaches
$x$. In this case $\beta$ is in the boundary of
a lift annulus and $l$ converges to the positive ideal
point in $L \cup \pin L$ and so $\Psi(l)$ converges to $L_+$ 
again.

The remaining case is that $\Theta(l)$ escapes in $\Theta(L)$.
Then as seen in $L \cup \pin L$ the ray $l$ converges
to the positive ideal point $p$ of all flow lines in $L$.
Hence $\Psi(l)$ converges to $\kappa(p) = L_+$.
Let $\Psi(v)$ be the limit point in any case.
Similarly if $l$ is a ray of $\wluf$.



Every point in $r$ it is $2 \delta_0$ close
to a point in $l$ in $F$, hence the limit of $\Psi(r)$ in
$\mi \cup \si$ is the same as that of
$l$. If $l'$ is another ray 
of $\wlsf$ or $\wluf$ converging to $v$,
then it will have
points boundedly close to $r$ which escape in 
$l'$ and therefore $\Psi(l')$ 
has the same ideal point
in $\si$. 
%Similarly if the point is an ideal
%point of leaves of $\wluf$. 
Therefore $\Psi(v)$ is well defined.

This finishes the construction of the extension
of $\Psi$ to $\pin F$.

\vskip .1in
\noindent
{\underline {Proof of continuity of the extension}} $-$

\vskip .1in
\noindent
{\underline {Case 1}} $-$  $v$ is not an ideal point of 
a ray in $\wlsf$ or $\wluf$.

Let $r$ be a geodesic ray in $F$ with ideal point $v$.
Recall the extension construction. 
There are $l_i$ in $\wlsf$ shrinking to $v$
in $F \cup \pin F$ and similarly $b_i$ in $\wluf$, 
assumed to be nested with the $l_i$.
Let $\{ l^*_i \}$ define a neighborhood basis of $v$ in
$F \cup \pin F$.
Let $L_i$ in $\wls$ with 
$l_i \subset L_i$, and $b_i \subset B_i \in \wlu$ 
as in the construction
case 1. Then as seen in the construction, the 
$L_i, B_i$ escape in $\mi$.
Let $U_i$ be the component of $F - l_i$ containing a subray of
$r$ and $V_i$ the component of $\mi - L_i$ containing $U_i$.
Notice that $\Psi(U_i) \subset V_i$.
Let now $z$ in $\overline U_i$ with the closure taken in 
$F \cup \pin F$ and $\overline V_i$ the closure of $V_i$
in $\mi \cup \si$.
Then $\overline U_i$ is a neighborhood of $v$ in $F \cup  \pin F$.
If $z$ is in $\Psi(\overline U_i)$ then using either of the
constructions in the extension part shows that $z$ is
a limit of points in $\Psi(U_i) \subset V_i$.
%If $z$ is in $V_i$ then its flow line has both ideal
%points in $\overline V_i \cap \si$, which has diameter
%decreasing to $0$ with $i$.
%A geodesic of $\mi$ with these ideal points
%is in a set of $\mi \cup \si$ of small visual diameter.
%Then since the orbits of $\wwp$ are uniform quasigeodesics
%they are a bounded distance from these geodesics. This shows
As seen in the construction arguments
the diameter of $\overline V_i$ in the visual distance
is converging to $0$. Hence we obtain continuity of $\Psi$ at
$v$.
This finishes the proof in this case.

\vskip .1in
\noindent
{\underline {Case 2}} $-$ $v$ is an ideal point of 
a ray of $\wlsf$ or $\wluf$.

This case is considerably more complicated, with several possibilities.

\vskip .1in
\noindent
{\underline {Case 2.1}} $-$
$v$ is an  ideal point of $\wlsf$ but not of $\wluf$
(or vice versa).

Suppose the first option occurs.
There is $l$ ray in $\wlsf$ with ideal 
point $v$. 
We may assume that $l$ is not in a leaf of $\wlsf$ with
same ideal points. Otherwise we can choose $l$ to be one
of the boundary leaves of the corresponding spike region.
Since $v$ is not an ideal point of $\wluf$,
there are $g_i$ line leaves
in $\wluf$ defining a basis neighborhood system at $v$.
Let $g_i$ be contained in $G_i$ leaves of $\wlu$.
Let $L$ in $\wls$ containing $l$.
If $G_i$ escapes in $\mi$ as $i \rightarrow \infty$, then as
seen in case 1, we are done.
Let then $G_i$ converge to the finite set of leaves

$${\cal V} \ =  \ H_1 \cup H_2 .... \cup H_m \ \ \ {\rm leaves
\ of } \ \ \wlu$$

\noindent
We can assume that $G_i \cap l \not = \emptyset$ for all $i$,
$G_i$ is non singular and the sequence $\{ G_i \}, i \in {\bf N}$
is nested.


\vskip .1in
\noindent
{\underline {Case 2.1.1}} $-$ Suppose that $L$ intersects
${\cal V}$, say $L \cap H_1 \not = \emptyset$.

Then $l$ escapes down as $\Theta(l)$ 
approaches $\Theta(L \cap H_1)$.
Otherwise $L \cap H_1$ is in the boundary of a lift annulus
$A$ and $l$ has a subray contained in this lift annulus.
But then $A$ is also contained in the unstable leaf $\wu(L \cap H_1)$
and so $G_i$ cannot intersect $l$, contradiction.
As $l$ escapes down in $L$, then
the ideal point of $\Psi(l)$ is
$(L \cap H_1)_-$ which is equal to $(H_1)_-$, the negative
ideal point of $H_1$. 

Since the values of $\Psi(p)$ for $p$ in $\pin F$ are obtained
as limits of values in $\Psi(F)$, then we only need to show that
if $z_k$ is in $F$ and $z_k$ converges to $p$ as $k \rightarrow \infty$,
then $\Psi(z_k)$ converges to $\Psi(p)$. Suppose this is not
the case.

%\blankfig{2.1}{1.1}{a. The leaves $\{ g_t \}$ converge
%to $v$ in $F \cup \pin F$, b. $L$ intersects $H_1$, which
%is in the limit of the $G_t$.}

By taking a subray if necessary, we may assume that $l$ 
does not intersect a lift annulus and hence it is transverse
to the unstable foliation $\wluf$ in $F$.
Parametrize the leaves of $\wluf$
intersected by $l$ as $\{ g_t, \  t \in \rrrr_+ \}$,
contained in $G_t \in \wlu$
(by an abuse of notation think of the $G_i$ as
a discrete subcollection of the $G_t, t \in \rrrr_+$). Let 

$${\cal U}  \ = \ \bigcup_{t>0} G_t$$

\noindent
No $g_t$ (or leaf of $\wluf$) 
has ideal point $v$ in $\pin F$.
As $\{ g_i \}, i \in {\bf N}$ converges to $v$ in $F \cup \pin F$, then
$g_t$ escapes compact sets in $F$ as
$t \rightarrow \infty$ and the ideal points of $g_t$ converge
to $v$ on either side of $v$. 
%If ideal points do not
%converge to $v$ then since ideal points of leaves of $\wluf$
%are dense in $\pin F$, there will be leaf $g$ in the
%limit of the $g_t$. 
%%If there is a leaf $g$ in the limit of the $g_t$, then
%Then since $\pi_1(M)$ is negatively
%curved there can only be finitely many leaves in
%the limit and consecutive leaves share an ideal
%point, because of the denseness again. It would then
%follow that some limit leaf has to have ideal point $v$,
%contradiction.
%It now follows that 
Up to subsequence assume that all of the elements
of the sequence $\{ z_k \}$ 
are either entirely contained
in ${\cal U}$ or disjoint from ${\cal U}$.

\vskip .1in
\noindent
{\underline {Situation 1}} $-$ Suppose that $z_k$ is not in ${\cal U}$ 
for any $k$.

Since $z_k$ is very close to $v$ in the compactification
$F \cup \pin F$
and $g_t$ converges to $v$ in $F \cup \pin F$ when $t \rightarrow \infty$,
then there are $t, s$
with $z_k$ between $g_t$ and $g_s$ (in $F$). Notice $z_k$ is not in
any of them. Now there is a unique time $t_k$ so that exactly at that
time $\Psi(z_k)$ switches from being in one side of $G_t$  in $\mi$
to the other
(equivalently compare the $z$ and $g_t$ in $F$).
In particular, either there is a line leaf $L_{t_k}$
of $G_{t_k}$ which separates $\Psi(z_k)$ from all the other 
$G_t$, see fig. \ref{sepa}, a, or there is a leaf $L_{t_k}$
non separated from $G_{t_k}$ with $\Psi(z_k)$ either in $L_{t_k}$
or $L_{t_k}$ separates $\Psi(z_k)$ from all $G_t$,
see fig. \ref{sepa}, b. This can be seen
in the leaf space of $\wlu$, which is a non Hausdorff tree 
\cite{Fe8,Ga-Ka,Ro-St}.


\begin{figure}
\centeredepsfbox{gm13.eps}
\caption{
a. Line leaf separating points, b. Non separated
leaf separating points.}
\label{sepa}
\end{figure}



\vskip .1in
\noindent
{\underline {Claim}} $-$ In the Gromov-Hausdorff topology of
closed sets of 
$\mi \cup \si$, the sets $\overline L_{t_k}$ 
converge to $(H_1)_-$ as
$k \rightarrow \infty$.

%\noindent
%Proof of the claim $-$ 
If $L_{t_k}$ is a line leaf of 
$G_{t_k}$, then $(L_{t_k})_- = (G_{t_k})_-$. If
$L_{t_k}$ is not separated from $G_{t_k}$ then also
$(L_{t_k})_- = (G_{t_k})_-$. 
This is because there are $E_i$ leaves of $\wlu$ with
$E_i$ converging to $L_{t_k} \cup G_{t_k}$.
So there are $x_i, \ y_i$ in $E_i$ with 
$x_i \rightarrow x, \ y_i \rightarrow y$ and $x \in L_{t_k}, \
y \in G_{t_k}$. Then 


$$\eta_-(x_i) \ \rightarrow \eta_-(x) \ = \ \eta_-(L_{t_k}), \ \ \ 
\eta_-(y_i) \ \rightarrow \eta_-(y) \ = \ \eta_-(G_{t_k})
\ \ \ \ {\rm and} \ \ \eta_-(x_i) \ = \ \eta_-(y_i).$$


\noindent
The last equality occurs
 because $x_i, y_i$ are in the same unstable
leaf $E_i$.
Therefore $(L_{t_k})_-$
converges to $(H_1)_-$ when $k \rightarrow \infty$. 
Suppose that $\overline L_{t_k}$ does not 
converge to $(H_1)_-$ in $\mi \cup \si$.
Since

$$(L_{t_k})_- \ \ {\rm converges \ to} \ \ (H_1)_-,$$

\noindent
then lemma \ref{narrow} shows that $L_{t_k}$ does not
escape compact sets in $\mi$. So it limits to some $u$ in
$\mi$ and up to subsequence  we may assume there 
$u_k$ in $L_{t_k}$ with $u_k$ converging
to $u$.
The first possibility is that the $L_{t_k}$ are subsets of 
%Suppose there are $u_k$ points in
%$L_{t_k}$ with the sequence 
%$u_{t_k}$ not converging to $(H_1)_-$. Then up to subsequence
%assume $u_k$ converges to $u$ a point in $\si$
% distinct from $(H_1)_-$.
%Hence $\wwp_{\rrrr}(v_k)$ does not escape $\hhh$ (because
%of the quasigeodesic property). Hence up to another subsequence
%we obtain
%
%$$\wwp_{\rrrr}(v_k) \ \ {\rm converges \ to} \ \ \wwp_{\rrrr}(v)$$
%
%\noindent
%in the compact open topology of closed sets in $\hhh$.
%In the first case $v_k$ are in 
the leaves $G_{t_k}$. This implies that $\wwp_{\rrrr}(u)$
is in the limit of the sequence of leaves $G_{t_k}$ (in $\mi$), so
it is contained in ${\cal V}$.
The second possibility is $L_{t_k}$ non separated from $G_{t_k}$ so
$L_{t_k}$ is between $G_{t_{k-1}}$ and
$G_{t_{k+1}}$ hence $u$ is again in the limit of the $G_t$ so
$u$ is in ${\cal V}$.
%the same property for $v$ holds.
%holds. 
The leaves $H_j$ in ${\cal V}$ are non singular in the
side the $G_t$ are limiting on, 
%But since
%$v$ is in $H$ and it is non singular in that region, there
so there is a neighborhood of $u$ on that side of $H_j$ which has
no singularities hence the $u_k$ will be in ${\cal U}$ for
$k$ big enough. This contradicts the hypothesis
in this case.

This shows that 
$\overline L_{t_k}$ converges to $(H_1)_-$ 
in $\mi \cup \si$.
Also $L_{t_k}$ 
either contains $\Psi(z_k)$
or separates it from a base point in $\mi$. It follows that
$\Psi(z_k)$ converges to $(H_1)_-$, which is what we wanted to prove.
This finishes the analysis in situation 1.

\vskip .1in
\noindent
{\underline {Situation 2}} $-$ For all $k$ assume that $\Psi(z_k)$ is
in ${\cal U}$.

Let $t_k$ with $\Psi(z_k)$ in $G_{t_k}$,
hence $z_k$ is in $G_{t_k} \cap F = g_{t_k}$. 
Then $(\Psi(z_k))_- = (G_{t_k})_-$ converges to $(H_1)_-$.
We want to show that $\Psi(z_k)$ converges to $(H_1)_-$.
Otherwise there is $q$  in $\si$
different from $(H_1)_-$ and a subsequence, still denoted by
$\Psi(z_k)$ so that
$\Psi(z_k)$ converges
to $q$.
As in the claim of situation 1 above, 
$\wwp_{\rrrr}(z_k)$ does not escape compact sets
in $\mi$ and there is $z$ in $\mi$, so that up to another
subsequence, we may 
assume that $\wwp_{\rrrr}(\Psi(z_k))$ converges to $\wwp_{\rrrr}(z)$.
Since $\Psi(z_k)$ is in $G_{t_k}$ then $z$ is
in ${\cal V}$, say $z$ is in $H_j$.
Let $p = \Theta(z)$.
At this point notice that $F$ does not intersect any 
leaf $H_i$ in ${\cal V}$.
If it did, say in $w$ then $F$ intersects the nearby leaves
$G_t$ (for any $t$ big enough) near $w$. This would
imply $F \cap G_t = g_t$ 
does not escape compact sets in $F$,
% hence $g_t$
%does not converge to $p$, 
contradiction.
Therefore 
%as $\Theta(\Psi(z_k))$ are in $\Theta(F)$ and
%$\Theta(p)$ is not $\Theta(F)$
% and in addition $\Theta(H_j)$ does
%not intersect $\Theta(F)$ then: $\Theta(H_j)$ has points
$p = \Theta(z)$ is in $\partial \Theta(F)$.
Let $x_k$ in $g_{t_k} \cap l$.
Then
$\Theta(x_k)$ converges to a point in $\Theta(H_1 \cap L)$.
There are segments $b_k$ in 
$F \cap G_{t_k} = g_{t_k}$ from $x_k$
to $z_k$. Then $\Theta(b_k)$ converges to a ray in
$\Theta(H_1)$
and a ray in $\Theta(H_j) \subset \oou(p)$ and possibly
other unstable leaves. Then there is a ray in $\oou(p)$
contained in $\partial \Theta(F)$. This implies that
$F$ escapes down as $\Theta(F)$ approaches this ray of $\Theta(H_j)$.
Hence $\Psi(z_k)$ is getting closer to $z_-$ which is $(H_j)_-$,
which is also equal to $(H_1)_-$.
This is what we
wanted to prove anyway.

This finishes the proof of case 2.1.1,
that is, when $L$ intersects ${\cal V}$.

\begin{lemma}{}{}
Let $A$ in $\wlu$, $B$ in $\wls$ satisfying:
there are
$R_i$ leaves of $\wlu$ intersecting $B$ 
with $R_i$ converging to $A$
and $R_i \cap B$ escaping compact sets in $B$.
Then $A_-$ is equal to $B_+$.
\label{conperf}
\end{lemma}

\begin{proof}{}
Since $R_i$ converges to $A$ then $(R_i)_-$ converges
to $A_-$.
Also $R_i$ intersects $B$ so
$(R_i)_- = (R_i \cap B)_-$. As $R_i \cap B$ escapes compact
sets in $B$ then in the intrinsic geometry of $B$,
the $R_i \cap B$ converges
to the positive ideal point of $B$. This implies that
$(R_i \cap B)_-$ converges to $B_+$. This implies the result.
\end{proof}


\noindent
{\underline {Case 2.1.2}} $-$ $L$ does not intersect 
${\cal V}$.

Then $\Theta(l)$ escapes in $\Theta(L)$ and so $\Psi(l)$ converges to
$L_+$. By the previous lemma,
this is also equal to $(H_1)_-$. From this point on,
the proof is the same
as in case 2.1.1.
This finishes the proof of case 2.1.

\vskip .2in
\noindent
{\underline {Case 2.2}} $-$ $v$ is an ideal point of both $\wlsf$ 
and $\wluf$.

\vskip .1in
\noindent
{\underline {Case 2.2.1}} $-$ For any ray $l$ of 
$\wlsf$ and $e$ of $\wluf$
with $l_{\infty} = e_{\infty} = v$, then $l$ does not intersect
$e$. 


Let $l', e'$ be rays as above. 
We may assume that $l', e'$ do not have any singularities.
Parametrize the leaves of $\wlsf$
intersecting $e'$ as $\{ l_t, t \geq 0 \}$ 
where $l_t \cap e'$ converges to $v$
in $F \cup \pin F$
as $t$ converges to infinity.

%We will prove continuity of $\Psi$
%at $v$ from one side of $e'$. The other side is
%treated in the same fashion.
Since $l'$ limits on $v$ and is disjoint from $e'$, then $l'$ 
is on a side
defined by $e'$. We will prove continuity of
$\Psi$ at $v$ from the other side
of $e'$. 
The point $p_t = l_t \cap e'$ disconnects $l_t$. For simplicity
we only consider those $l_t$ with $l_t \subset L_t \in \wls$ and
$L_t$ non singular. Let $l^1_t$ be the component of $(l_t - p_t)$ in 
the $e'$ side containing $l'$ 
union with $p_t$. Let $l^2_t$ be the other component
of $(l_t - p_t)$ union with $p_t$, see fig. \ref{onsid}, a.

The $l^1_t$ are rays (here we use $L_t$ non singular - but this
is just a technicality) and $(l^1_t)_{\infty}$ are not equal $v$ by
hypothesis. They cannot escape compact sets of $F$ since
$l'$ with ideal point $v$ is on that side of $e'$.
%Also they are bounded by $l'$. 
Hence as $t$ converges
to infinity $l^1_t$ converges to a leaf $l$ of $\wlsf$
with a ray (also denoted by $l$) with ideal point $v$ and
maybe some other leaves as well. The leaf
$l$ either shares a subray with $l'$ or separates $l'$ from $e'$
Let $e' \subset E$ leaf of $\wlu$ and $l \subset L$, leaf of $\wls$.


\vskip .1in
\noindent
{\underline {Case 2.2.1.1}} $-$ $l^2_t$ escapes in $F$
as $t \rightarrow \infty$.

Let $b_t$ be the ideal point of $l^2_t$. Then $b_t \not = v$.
Let $L^2_t$ be the union of $\wwp_{\rrrr}(p_t)$ and the component
of $L_t - \wwp_{\rrrr}(p_t)$ containing $l^2_t$. If $L^2_t$ escapes
in $\mi$, then the arguments in case 1 show 
continuity of 
$\Psi$ at $v$ in
the side of $e'$ not containing $l'$.

Now assume that $L^2_t$ converges to $R_1 \cup ... \cup R_m$
leaves of $\wls$
with union ${\cal R}$.
Notice $F$ may intersect some of these leaves or not.
If $\Theta(\Psi(p_t))$ does not escape in $\Theta(E')$,
then one of the $R_i$, call it $R_1$, is a leaf
intersecting $E'$.
As seen in the arguments for case 2.1.1, $F$ escapes
up in this direction so $\Psi(p_t)$ converges to
$(R_1)_+$.
If $\Theta(\Psi(p_t))$ escapes in $\Theta(E')$, then
lemma \ref{conperf} shows that $\Psi(p_t)$ also
converges to $(R_1)_+$.
This is equal to $(R_j)_+$ for any $j$.

Suppose there are $t_k \rightarrow \infty$ and 
$z_k$ in $l^2_{t_k}$ with $\Psi(z_k)$ not converging
to $(R_1)_+$. 
Here there is no need to assume that $L_{t_k}$ is non singular.
Then there $q$ in $\si$, \ $q \not = (R_1)_+$
and a subsequence still denoted by $\Psi(z_k)$,
so that $\Psi(z_k)$ converges to $q$.
As before there is $z$ in $\mi$ so that up to another 
subsequence, still denoted by $\wwp_{\rrrr}(z_k)$,
then $\wwp_{\rrrr}(z_k)$  converges
to $\wwp_{\rrrr}(z)$ and hence $z$ is in ${\cal R}$, say in
$R_i$.
%As in case ???
Then $\wwp_{\rrrr}(z_k)$ are near 
$\wwp_{\rrrr}(z)$ and since a ray of $\Theta(R_i)$ is in
$\partial \Theta(F)$, then this is stable boundary. 
So $F$ escapes up as $\Theta(F)$ approaches $\Theta(z)$ and hence 
$\Psi(z_k)$ converges to $(R_i)_+$. 
This is equal to $(R_1)_+$.
The arguments of Case 2.1.1, situation 1 then show
continuity of $\Psi$ at $v$ on this side
of $e'$. This finishes the analysis of case 2.2.1.1.



\begin{figure}
\centeredepsfbox{gm14.eps}
\caption{
a. Convergence on one side,
b. Case 2.2.1.2 - intersection of leaves.}
\label{onsid}
\end{figure}




\vskip .1in
\noindent
{\underline {Case 2.2.1.2}} $-$ The \ $l^2_t$ \ limit to $r$ in $F$
as $t \rightarrow \infty$.

Choose the leaf $r$ with a ray which has ideal point $v$.
Then the leaves $r, l$ are not separated from each other in the leaf
space of $\wlsf$. 
%Proposition \ref{band} shows that the region bounded by these 
Since $r, l$ have ideal point $v$ and there is no leaf
of $\wlsf$ non separated from $r, l$ and between them,
proposition \ref{nh} shows that the region bounded by these
rays of $r, l$ with ideal point $v$ projects in $M$ to a set
asymptotic to a Reeb annulus.
It follows that in $F$ this region is a bounded 
distance from
a geodesic ray with ideal point $v$. 
Now we restart
the process with the ray $r$ of $\wlsf$ instead of $e'$ of $\wluf$.
Let $\{ b_t, t \geq 0 \}$ be a parametrization of the 
leaves of $\wluf$ 
through the corresponding points $x_t$ of $r$.
%see fig. \ref{nonsep1}, a.
If the  components of $(b_t - x_t)$ on the side opposite
of $e'$ escapes compact sets in $F$, then the analysis
of case 2.2.1.1 shows continuity of $\Psi$ at $v$ in that
side of $r$. Since $r$ and $e'$ are a bounded distance
from each other in $F$, this shows continuity of $\Psi$ at
$v$ on that side of $e'$.

Otherwise this process keeps being repeated.
Let $A_0 = L$, $A_1$ be the leaf of $\wls$ containing $r$.
If the process above does not stop, we keep producing $A_i$ in $\wls$,
so that they all disjoint and $A_i$ is non separated from
$A_{i+1}$. By theorem \ref{theb} 
up to covering translations there are only finitely many
leaves of $\wls$ which are not separated from some other leaf of
$\wls$.
There is then $m > n$ and $h$ covering translation with
$h(A_n) = A_m$. Let $f$ be the generator of the joint
stabilizer of $A_0, A_1$. 
This is non trivial by theorem \ref{theb}.
Then $f$ preserves all the prongs
of $A_1$ and therefore leaves invariant all the $A_i$.
Hence $h^{-1} f h(A_n) = A_n$ and so $h^{-1} f h = f^a$ for
some integer $a$. This implies there is a ${\bf Z} \oplus {\bf Z}$
in $\pi_1(M)$, see detailed arguments in \cite{Fe8}.
This is a contradiction.

There is then a last leaf $l_y$ (of $\wlsf$ or $\wluf$)
obtained from this process. The arguments of case 2.2.1.1 show
continuity of $\Psi$ at $v$ on the other side of $l_y$.
The region between $e'$ and $l_y$ is composed of a finite
union of regions between non separated rays of $\wlsf$ or
$\wluf$. They are all a bounded distance from a geodesic
ray with ideal point $v$, so the whole region also
satisfies this property.  It follows that this region
can only limit in $\Psi(v)$ as well and this proves
continuity of $\Psi$ at $v$ in that side of $e'$.


An entirely similar analysis shows continuity of $\Psi$ at
$v$ from the side of $l'$ not containing $e'$.

\vskip .1in

What remains to be analysed is the region of $F$ 
{\underline {between}} the rays $l'$ and $e'$.
Consider first the case that there is pair of non separated
leaves in the chain from $l'$ to $e'$.
Then as seen before the region between $l'$ 
and  $e'$
%there is non Hausdorfness involved, this region
is a bounded distance (the bound is not uniform) from a geodesic
ray with ideal point $v$. 
This is not the case a priori if
there is no non Hausdorfness involved. 
In this case the region between $l'$ and $e'$ may not have
bounded thickness in $F$ and hence it is unclear whether
its image under $\Psi$ can only limit in $\Psi(v)$.
We analyse this case now.

In this last case 
parametrize the leaves of $\wluf$ intersecting the ray 
$l$ of $\wlsf$ as \ $\{ e_t \ | \ t \geq 0 \}$.
Since $l_t$ converges to $l$, then for big enough
$t$, the leaves $l_t, e_t$ intersect $-$ let $u_t$ be
their intersection point, see fig. \ref{onsid}, b.
Now define $l^*_t$ to be
the component of $l_t - u_t$  intersecting $e$ and $e^*_t$ 
the component of
$e_t - u_t$ intersecting $l$. 
Since $e'$ is on that side of $l$, the $e_t$ cannot escape
and converge to a leaf $e$ of $\wluf$ with an ideal point
$v$. Let $e \subset E$  leaf of $\wlu$.

Recall that $L_t$ is the leaf of $\wls$ containing $l^*_t$
and similarly let $E_t$ be the leaf of $\wlu$ containing $e_t$.
Let $L^*_t$ be the component
of $L_t - \wwp_{\rrrr}(u_t)$ containing $l^*_t$
and similarly define $E^*_t$.
In this remaining case the $l^*_t$ escape in $F$ and
so do the $e^*_t$. Hence
$\mu_t \ = \ l^*_t \cup \{ u_t \} \cup e^*_t$
defines a shrinking neighborhood system of
$v$ in $F \cup \pin F$. Consider the set

$$B_t \ = \ L^*_t \cup \wwr(u_t) \cup E^*_t$$

We want to show that $\overline B_t$ converges
to $L_+$ in the topology of closed sets of $\mi \cup \si$.

First consider $L^*_t \cap E$ which intersects $F$
in $(l^*_t \cap e)$. If $L^*_t \cap E$ does not
escape compact sets in $E$ then it limits to an orbit
$\gamma$ contained in a leaf $H$ of $\wls$.
Then $L, H$ are not separated from each other.
But for $t$ big enough then $E_t$ is near enough
$E$ and will intersect $H$ as well. This contradicts
$E_t \cap L$ is not empty and $L, H$ non separated.
Hence $L^*_t \cap E$ escapes in $E$ and similarly
$E^*_t \cap L$ escapes in $L$. 
Hence $L, E$ form a perfect fit.
This implies that $L_+ = E_-$.
Also $\Psi(e)$ limits to $E_-$ and $\Psi(l)$ limits
to $L_+ = E_-$.

The set $\overline L^*_t$ contains 
$(L^*_t \cap E)_+$ and this converges to $E_-$
when $t \rightarrow \infty$.
This is because $(L^*_t \cap E)$ escapes in $E$.
If $\overline L^*_t$ does
not converge to $E_-$ in $\mi \cup \si$, then
we find $t_k \rightarrow \infty$ and $x_k \in
L^*_{t_k}$ with $x_k$ converging to $x$ not equal to $E_-$.
Since $(x_k)_+ = (L_{t_k})_+$ converges to $E_-$,
then up to subsequence assume
$\wwr(x_k)$ converges to $\wwr(z)$ for some $z$ in $\mi$.
Then $z$ is in a leaf $H$ of $\wls$ which is non
separated from $L$.
%, see fig. \ref{fini}.

%\blankfig{fini}{1.1}{Non separated leaves in $\mi$.}

%Since all the $x_k$ are in the component of $\mi - E_t$
%containing $E$ then $H$ is separated from $L$ by $E$, see
%fig. \ref{fini}.
The leaf $H$ does not intersect $F$, because $l^*_t$
escapes in $F$ by hypothesis in this final situation.
It follows that $\Theta(H)$ has a ray contained
in $\partial \Theta(F)$ and so this is stable boundary
of $\Theta(F)$. Hence $F$ escapes up as $\Theta(F)$
approaches $\Theta(H)$ and consequently 
$\Psi(x_k)$ limits to $H_+ = L_+ - = E_-$
$-$ which is what we wanted anyway.
This shows that $\overline L^*_t$ converges
to $E_-$ in $\mi \cup \si$.

Analysing the sets $E^*_t$ in the same manner we obtain
that $\overline E^*_t$ converges to $L_+$ as $t \rightarrow \infty$
as well.
This implies that $\overline B_t$ converges to 
$L_+ = \Psi(v)$. Since $B_t \cap F = \mu_t$ and the
$\mu_t$ define a neighborhood basis of $v$ in $F \cup \pin F$,
this shows continuity of $\Psi$ at $v$. 
This finishes the proof of case 2.2.1.2 and hence of
case 2.2.1.




\vskip .2in
\noindent
{\underline {Case 2.2.2}} $-$ There are rays $l$ of $\wlsf$ and
$e$ of $\wluf$ starting at $u_0$ and having the ideal point $v$.

We will first prove continuity on the side of $e$ not containing
a subray of $l$. There will be an iteration of steps.
Before we start the analysis we want to get rid of some
problems as described now. Suppose that there are $\alpha_0, 
\beta_0$ leaves of $\wlsf$ (or leaves of $\wluf$)
which have non separated rays converging to $v$ in $\pin F$
and on that side of $e$.
Suppose there are infinitely many of these on that side of
$e$. Let them be $\alpha_i, \beta_i$ and $G_i$ in
$\wls$ containing $\alpha_i$. Each region $B$ between $\alpha_0$
and any $\alpha_i$ is a bounded distance from a geodesic
ray in $F$ with ideal point $v$. The image $\Psi(B)$ then
can only limit in $\Psi(v)$.
If the $G_i$ do not escape in $\mi$ then they converge
to a leaf $G$ of $\wls$.
Let $A$ be an unstable leaf intersecting $G$ tranversely.
For $i$ big enough then $A$ intersects $G_i$ transversely,
which is impossible, as it would intersect
$\alpha_i$ and $\beta_i$ and these are not separated.
Hence the 
the $G_i$ escapes in $\mi$. Then  as seen in case 1,
there is continuity of $\Psi$ at $v$ in that side of $\alpha_1$.

Another situation is when there are leaves $\alpha_i$ in
that side of $e$ with two rays with ideal point $v$.
Then they are in the interior of a spike region $B$ with
one boundary $g$ with ideal point $v$. If there are infinitely
many of these, where none of the $\alpha_i$ are
nested with each other, then
 let $G_i$ in $\wls$ containing $\alpha_i$.
As in the previous paragraph,
the $G_i$ have to escape in $\mi$ and we have continuity
in that side of $\alpha_1$.

Therefore we can assume there are only finitely many
occurrences of spike regions or non separated leaves
with ideal point on this side of $e$. 
If there is any of these let $e_0$ be the last ray
in that side coming from such occurrences. 
Otherwise let 
$e_0$ be the ray given $e$ by the hypothesis in this case.
For simplicity assume that $e_0$ is a ray in $\wluf$,
the other case being similar. Let $e_0 \subset E_0 \in \wlu$.


Parametrize the ray of $e_0$ as 
\ $\{ p_t \ | \ t \geq 0 \}$ \
with $p_t$ converging to $v$ as $t \rightarrow \infty$. Let $l_t$ be
the leaf of $\wlsf$ through $p_t$ and $L_t$ in $\wls$
with $l_t \subset L_t$. If $L_t$ escapes $\mi$ as
$t \rightarrow \infty$ then as seen before we have continuity 
of $\Psi$ at
$v$ in that side of $e_0$.
So now suppose that $L_t$ converges to $A_1 \cup .... A_m$,
leaves of $\wls$. 
%Then $F$ escapes up as $\Theta(F)$ approaches $\Theta(A_i)$
%for any $i$.
This case is considerably more involved, with several 
possibilities.

\vskip .1in
\noindent
{\underline {Claim}} $-$ $\Psi(e_0)$ converges to $(A_i)_+$ 
(notice the  $(A_i)_+, 1 \leq i \leq m$ are all equal).

If $E_0$ intersects some $A_i$, say $A_1$, then
as seen in case 2.1.1,
$F$ escapes positively along $\Psi(e_0)$ as $\Theta(F)$
approaches $A_1$. This implies that
$\Psi(e_0)$ converges to $(E_0 \cap A_1) _+ = (A_1)_+$.
If $E_0$ does not intersect any 
$A_i$ then $\Psi(e_0)$ converges 
to $(E_0)_- = (A_1)_+$. This proves the claim.

Let $l^1_t$ be the component of $(l_t - p_t)$ in the side
of $e_0$ we are considering.
We are really interested
in the behavior for $t \rightarrow \infty$, so we may
assume $p_t$ is not singular and there is only one
such component.

Suppose first that no $l^1_t$ has a ray with
ideal point $v$ and that
$l^1_t$ escapes in $F$ as $t \rightarrow \infty$.
In this case it is easy to show continuity 
of $\Psi$ at $v$ and in this side of $e_0$:
Suppose there are $x_i$ in $l^1_{t_i}$ with $t_i \rightarrow \infty$
and $\Psi(x_i) \not \rightarrow (A_i)_-$.
Since $(x_i)_+$ converges to $(A_i)_+$ then up to subsequence
assume that $(x_i)_- \rightarrow b \not = (A_i)_+$.
Up to subsequence $\wwr(x_i) \rightarrow \wwr(x)$. Then $x$
is in some $A_i$ say $x \in A_2$. But $F$ escapes positively
as 
$\Theta(F)$ approaches $\Theta(A_2)$,
so $\Psi(x_i) \rightarrow (A_i)_+$,  as we wanted.
Then as in case 2.1.1 this implies continuity.


There are 2 other options: 1) There is no $t$ with $l^1_t$
with an ideal point $v$ and $l^1_t$ does not escape in $F$; and
2) There is $t$ with $l^1_t$ having ideal point $v$.
These two options interact and intercalate in appearance
as explained below:


\vskip .1in
\noindent
{\underline {Situation 1}} $-$  There is no $t$ with $l^1_t$
with ideal point $v$ and $l^1_t$ does not escape in $F$.

%Otherwise the limit leaf would be in the interior
%of a spike region and so would $l_t$ for $t$ big,
%contradiction to identified ideal points.
There could be several leaves of $\wlsf$ in the limit of $l^1_t$ 
as $t \rightarrow \infty$ 
but there is a single leaf, call it $g$ with ideal point $v$.
If there is more than one such leaf with ideal point $v$, then
there would have to be one with
two rays with ideal point $v$. This leaf would be in a spike region
and it is separated from any other leaf in $\wlsf$,
contradiction.
Let $g$ be contained in a leaf 
 $G$ of $\wls$.
%The arguments of case 2.2.1.2 show that if $B$ is the region in $F$ 
%between $e_0$ and $g$,  then $\Psi(B)$ can 
%only limit at $\Psi(v) = (A_i)_+$.


\begin{figure}
\centeredepsfbox{gm15.eps}
\caption{
Some limits in $F$, b. The picture in $\mi$.}
\label{remo}
\end{figure}


Parametrize the ray $g$ as \ $\{ q_t \ | \ t \geq 0 \}$,
with $q_t \rightarrow v$ as $t \rightarrow \infty$.
Let $s_t$ be the unstable leaf of $\wluf$ through
$q_t$. 
Let $s^1_t$ be the component of $(s_t - q_t)$ on the
side of $g$ opposite to $e_0$ and $s^2_t$ the other
component. Then $s^2_t$ cannot have ideal point
$v$: for $t$ big enough it intersects $l^1_t$,
see fig. \ref{remo}, a.
Then $s^2_t$ converges to $e_0$.
By hypothesis there are no more occurrences of non separated
leaves of $\wlsf$
with ideal point $v$ on that side of $e_0$,
which implies that $s^1_t$ 
cannot limit to a leaf of $\wluf$ at $t \rightarrow \infty$
(it would distinct but non separated from $e_0$).
Hence the $s^1_t$ have to escape compact sets
in $F$. 
If $s^1_t$ does not have
an ideal point at $v$ for any $t$, then 
the previous analysis shows continuity of $\Psi$ at $v$ in
that side of $g$.
As in case 2.2.1.2
if $B$ is the region between $g$ and $e_0$ then
$\Psi(B)$ can only limit in $\Psi(v)$.

Hence assume 
there is some $t_0$ so that $s^1_{t_0}$ has ideal point $v$,
see fig. \ref{remo}, a.
Then for $t$ bigger than $t_0$ all ideal points of $s^1_t$
are $v$.
Let $s^1_{t_0}$ be contained in a leaf $S$
of $\wlu$ and $s_t$ contained in $S_t$ leaf of $\wlu$.
Since

$$l^1_t \ \rightarrow \ g, \ \ \ s^2_t \ \rightarrow \ e_0
\ \ \ \ {\rm when } \ \ t \rightarrow \infty,$$


$${\rm then} \ \ L_t \ \rightarrow \ G, \ \ \ 
S_t \ \rightarrow \ E_0, \ \ \ {\rm when} \ \ 
t \rightarrow \infty.$$

\noindent
It follows that $E_0, G$ form a perfect fit,
see fig. \ref{remo}, b.
Hence $(E_0)_- = G_+$.
If $\Theta(s^1_{t_0})$ is a ray in $\Theta(S)$ then
$\Psi(s^1_{t_0})$ converges to $S_-$.
But $\Psi(s^1_{t_0})$ also converges to 

$$\Psi(v)  \ = \ (E_0)_- \ = \ G_+ \ = \ (G \cap S)_+.$$

\noindent
Let $\gamma_0 = G \cap S$, an orbit of $\wwp$ in $G$.
The above equations imply that 

$$(\gamma_0)_+ \ = \ (G \cap S)_+  \ =  \
\Psi(v) \ = \ S_- \ = \ 
(\gamma_0)_-,$$


\noindent
which is a contradiction.
Hence $\Theta(s^1_t)$ is not a ray and has an endpoint
$x_1$ in $\Theta(S)$. Let $\gamma_1 = x_1 \times \rrrr$.
Let $H = \wls(\gamma_1)$. 
But $F$ does not intersect $H$.
If $F$ escapes down as $\Theta(F)$ approaches $x_1$, then
$\Psi(v) = (\gamma_1)_-$. 
But then

$$(\gamma_0)_- \ = \ (\gamma_1)_- \ = \ \Psi(v) 
\ = \ (\gamma_0)_+$$

\noindent
contradiction.
This implies that $F$ escapes up as $\Theta(F)$ approaches
$x_1$. 
Hence $\Theta(H)$ has a ray in $\partial \Theta(F)$.
%Then $S$ intersects $H$ and for $t$ near $t_0$, so does $S_t$.
%This implies that $\Theta(H)$ has a ray in $\partial \Theta(F)$.
Therefore $\Psi(s^1_t)$ limits to $(\gamma_1)_+$.
This implies that $(\gamma_0)_+ = (\gamma_1)_+$,
where $\gamma_0, \gamma_1$ are distinct orbits
of $\wwp$ in the {\underline {same}} unstable leaf $S$.
This is dealt with by the following theorem proved
in \cite{Fe5}:

\begin{theorem}{}{(\cite{Fe5})}
Let $\Phi$ be a quasigeodesic almost pseudo-Anosov flow in $M^3$ with
$\pi_1(M)$ negatively curved. Suppose there is
an unstable leaf $V$ of $\wlu$ and different orbits
$\beta_0, \beta_1$ in $V$ with $(\beta_0)_+ = (\beta_1)_+$.
Then $C_0 = \wls(\beta_0),  \ C_1 = \wls(\beta_1)$ are both periodic,
invariant under a nontrivial covering translation $f$,
and the periodic orbits in $C_0, C_1$ are connected
by an even chain of lozenges all intersecting $V$.
\label{ident}
\end{theorem}

\noindent
{\bf {Remark}} $-$
This result is case 2 of theorem 5.7 of \cite{Fe5}.
In that article the proof is done for quasigeodesic
Anosov flows in $M^3$ with $\pi_1(M)$ negatively
curved. The proof goes verbatin to the case of
pseudo-Anosov flows. The singularities make no
difference. By the blow up operation, the same holds
for almost pseudo-Anosov flows.
\vskip .1in


%Assume first that none of the $r_t$ has limit $p$.
%In the joint figure in $\mi$, (figure \ref{remo}, b) we see
%a continuun of leaves $g_0$ to $r_t$ which are all escaping to $p$.
%There is a last leaf $G'$. Now the other ideal point
%of $g_0$ cannot be $p$ or else it would be in the
%non separated case.
%If some $l_t$ is part of a leaf of $\wlsf$ with 
%2 ideal points identified, then this $l_t$ is inside a strip ${\cal S}$
%which is a bounded distance from a  geodesic ray. 
%Hence the whole
%half of the strip converges to $(H_i)_+$. Consider a parametrization
%$\{ q_t \ | \ t \geq 0 \}$ for the outer ray $x$ of the strip
%and let $x_t$ be the unstable leaves of $\wluf$ through $q_t$.
%If some has identifications, we produce another pair of
%non separated leaves. As seen before this process has to stop.
%For simplicity assume the last boundary leaf is a stable one
%and call it $s_0$ also.
%
%\blankfig{quilt}{1.1}{Escaping and non separated leaves.}
%
%\vskip .1in
%Suppose first that none of the $r_t$'s has endpoints $p$.
%If the $r_t$ escape in $F$ then as seen in the
%claim, we obtain continuity in this side of $p$. If on the
%other hand we obtain $r_t$ converging to $r'$, then again
%we obtain non separated leaves $R'$ and $G'$ in $\wlu$.
%Again this produces a non Hausdorff strip that in $M$
%is asymptotic to a Reeb annulus in a leaf of $\fol$. Since
%any two such strips are a bounded distance apart, the same
%happens to their limits, so we produce non Hausdorff strips in
%the same leaf of $\fol$. 
%
%Since there are only finitely many non Hausdorff leaves of $\wlu$
%up to covering translations, they will project to the same leaf
%in $M$. This produces a ${\bf Z} \oplus {\bf Z}$ in $\pi_1(M)$
%as seen before and this is impossible.
%
%So now assume that some $r_0$ has ideal point $p$. Consider
%$s_t$, the set of leaves of $\wls$ intersecting $r_{\infty}$.
%One of the endopoints is $p$ and the other endpoint is also $p$
%will produce non Hausdorfwless. So assume the other endpoint
%is not $p$.
%
%Let $s_t \subset S_t \in \wls$.
%Notice that $R_b$ intersects $E_0$ and
%
%$$(R_b \cap E_0)_+ \ = \ (E_0)_+ \ = \ (H^*)_-.$$
%
%
%\noindent
%If $\Theta(r_b)$ escapes in $\Theta(R_b)$ then $r_b$ converges
%to $(R_b)_-$. But this is equal to $(R_b \cap E_0)_+$,
%contradiction.
%So $\Theta(r_b)$ limits to a leaf $U_0$
%which is a stable leaf, so $r_b$ escapes going up and
%$r_b$ limits to
%$(R_b \cap E_0)_+$. Then
%
%$$(R_b \cap E_0)_+ \ = \ (R_b \cap V_0)_+$$
%
%\noindent
%This means that 2 distinct orbits in the same unstable leaf $R_b$
%have the same positive ideal point. This situation was analysed
%before.
%
%The region $B$ bounded $l_0$ and $e_0$ with ideal point
%$v$ does not have any interior singularity. If it did
%we would have a line leaf in $B$ and hence a leaf with 
%both endpoints at $v$. By proposition \ref{band},
%that leaf would be
%part of a spike region and the spike region 
%does not have any singularities in its interior.
%
%\begin{proposition}{}{}
%In the situation above, then $E_0, R_0$ are the sides of lozenges
%connected by a finite chain all intersecting $R_b$.
%\end{proposition}

The theorem implies that $G, H$ are in the boundary of a chain
of adjacent lozenges all intersecting $S$. The first lozenge,
call it ${\cal C}$ has one stable side contained in $G$
and an unstable side $D_1$ which makes a perfect fit with
$G$. Suppose first $D_1$ is in the side of $S$ opposite
to $E_0$, see fig. \ref{spread}, a. The other unstable side
of ${\cal C}$ is a leaf $D_2$ which intersects $G$ on the
other side of $S$. Hence $G$ is some $S_c$ with $c > t_0$.
Then $S_c \cap F = s_c$ \ is a leaf of $\wluf$ and
$\Psi(s_c)$ has ideal point $\Psi(v)$. Notice that
$\Theta(s_c)$ (which is contained in $\Theta(F)$) escapes
in $\Theta(F)$ $-$ otherwise it would produce stable/unstable
boundary in $\Theta(F)$ before it hits $\Theta(H)$
and $\Theta(F)$ could not limit on $\Theta(H)$, impossible.
Hence $\Psi(s_c)$ limits to $(S_c)_-$ which is equal to $\Psi(v)$.
Then

$$(S_c \cap G)_- \ = \ (S_c)_- \ = \ \Psi(v) \ = \ G_+$$

\noindent 
which contradicts the orbit $S_c \cap G$ being a
quasigeodesic.


%Then the forward ideal point is $(E_0)_+$ which is $(S)_-$, etc..
%If the chain of nonseparated leaves is not on the $H_*$ side,
%then there is $R_c$ with $c > b$ so that $R_c$ makes a perfect
%fit with $A$ one side of a lozenge. Then $r_c$ has ideal point
%$(R_c)_- = (R_c \cap E_0)_-$. This is a contradiction
%because the limit point is also $(R_b \cap E_0)_+ = (E_0)_+$.

It follows that the perfect fits with $G$ occurs in the $E$ side
of $S$, see fig. \ref{spread}, b.
Here $\Theta(H), \Theta(D_1)$ are
contained in the boundary of $\Theta(F)$. 
%As seen
%in case ???? any points
%in $F$ in the region between $e_0$ and $s_{t_0}$ can only
%limit to $(D_1)_- = H_+$.
We now look at the region $B$ in $F$ bounded by $s_{t_0} = S \cap F$
and $e_0 = E_0 \cap F$. 


\vskip .1in
\noindent
{\underline {Claim 1}} $-$ 
The image $\Psi(B)$ can only
limit in $\Psi(v)$. 


The region $\Psi(B)$ is contained in the region ${\cal E}$ of
$\mi$ which is bounded by $E, D_1$ (maybe other unstable leaves non
separated from $D_1$ as well), $H$ and $S$, see fig. \ref{spread}, b.
Notice that $F$ does not intersect $D_1$ or any leaf non separated from
$D_1$ which is beyond $D_1$. Otherwise $b_0 = (D_1 \cap F)$ is
contained in $B$ and non separated from $e_0$, so it would have
both ideal points $v$. Then it would be contained in the
interior of a spike region and could not be non separated
from another leaf $-$ impossible.
On the other hand since $\Theta(H)$ has a line leaf in
the stable boundary of 
$\Theta(F)$, then $\Theta(D_1)$ has a line
leaf in the unstable boundary of $\Theta(F)$.
Hence $F$ escapes down as $\Theta(F)$ approaches 
$\Theta(D_1)$.

Let $z_k$ in $B$ escaping in $F$ and hence
converging to $v$ in $\pin F$.
Suppose that $\Psi(z_k)$ does not converge to $\Psi(v)$.
Given that $z_k$ escapes $F$ and the structure of the region
${\cal E}$, it follows that up to subsequence
either $\wu(z_k)$ converges
to $D_1$ or $\ws(z_k)$ converges to $H$.
Suppose that $\ws(z_k)$ converges to $H$.
In that case $(z_k)_+$ converges to $H_+ = \Psi(v)$.
Then as seen before if $(z_k)_-$ does not converge
to $\Psi(v)$ we can assume up to subsequence $\wwr(z_k)$
converges to $\wwr(z)$. Then $z$ is in a leaf
non separated from $H$ and since $\Psi(z_k)$ has to
be in ${\cal E}$ then $z$ can only be in $H$.
As $F$ escapes up as $\Theta(F)$ approaches $\Theta(H)$ then
$\Psi(z_k)$ converges to $H_+ = \Psi(v)$.
The case $\wu(z_k)$ converges to $D_1$ leads to 
$\wwr(z_k)$ converging to $\wwr(z)$ with $z$ in
unstable leaf non separated from $D_1$. As $F$ escapes
down as $\Theta(F)$ approaches these unstable leaves,
then $\Psi(z_k)$ converges to $(D_1)_- = \Psi(v)$.
Since this works for any subsequence of $z_k$, then
$\Psi(z_k)$ has to converge to $\Psi(v)$ always.
This proves claim 1.


\begin{figure}
\centeredepsfbox{gm16.eps}
\caption{
Perfect fit
with $G$ in the side opposite to $E_0$,
b. Perfect fit in the $E$ side.}
\label{spread}
\end{figure}





\vskip .1in
Let $G_0 = G$.  Notice that $G$ is periodic and connected to $H$ 
by an even chain of lozenges. 
We consider the ray $s_{t_0} = S \cap F$ which has
ideal point $v$. Parametrize it as \ $\{ z_t \ | \ t \geq 0 \}$.
Let $y_t$ be the leaf of $\wlsf$ through $z_t$ and $y^1_t$ the
component of $(y_t - z_t)$ in the side opposite to $e_0$.
The ray $s_{t_0}$ has the same behavior as the original ray
$e_0$. Hence we obtain continuity in that side of $s_{t_0}$
unless $y^1_t$ converges to a leaf $\mu$ of $\wlsf$ with
ideal point $v$. Let $G_1$ in $\wls$ with $\mu \subset G_1$.
Then $G_1$ is non separated from $H$, see fig. \ref{spread}, b \
and therefore connected to it by a chain of lozenges.
It follows that $G_1$ is connected to $G_0$ by a chain
of lozenges.
As in the proof of claim 1, the region $B_1$ of $F$
between $e_0$ and $(F \cap G_1)$ has image
$\Psi(B_1)$ which can limit only in $\Psi(v)$.

We restart the process with $g_1 = G_1 \cap F$ 
instead of $g$. The leaves of $\wluf$ through
points of $g_1$ already converge to the unstable
leaf $(D_3 \cap F)$ of $\wluf$ \ ($D_3$ is depicted in fig.
\ref{spread}, b).
The leaf $(D_3 \cap F)$ cannot be non separated from any
other leaf of $\wluf$ in that side of $(D_3 \cap F)$.
It follows that 
the unstable leaves intersected by $g_1$ escape
in $F$. The only case to be analysed is that some of
these unstable leaves have ideal point $v$. This brings
the process exactly to the situation of some $s^1_t$
of $\wluf$ having ideal point $v$ as described before
(it was $s^1_{t_0}$). So this would produce 
$H_1$ of $\wls$ with similar properties as $H$.
This process can now be iterated.
As in claim 1 the region of $F$ between $g_i$ and $g_{i+1}$
maps to $\mi$ to a region which can only limit in $\Psi(v)$.

We show that this process has to stop. Otherwise produce
$G_i$ leaves of $\wls$ which are all connected to $G_0$ by
a chain of lozenges. The $G_i$ are all non separated from
some other leaf of $\wls$, Hence there are $G_i, G_j$ which
project to the same stable leaf in $M$.
There is a covering translation $h$ taking $G_i$ to $G_j$.
If $f$ is a generator of the isotropy group of $G_0$ leaving
all sectors invariant, then it leaves invariant
all lozenges in any chain starting in $G_0$ so leaves
invariant all the $G_i$. As before this leads to $h^{-1} f h = f^n$
for some $n$ in ${\bf Z}$ and to a ${\bf Z} \oplus {\bf Z}$ in
$\pi_1(M)$. This is disallowed.
Therefore the process finishes after say $j$ steps and we
obtain continuity of $\Psi$ at $v$ in that side of $g_j = G_j \cap F$.
As seen above the region between $s_{t_0}$ and $g_j$ maps by $\Psi$
into a region that can only limit in $\Psi(v)$. 
This proves continuity of $\Psi$ at $v$ in that side of $e_0$.
This finishes the analysis of situation 1.




\vskip .15in
\noindent
{\underline {Situation 2}} $-$ There is $l^1_{t_0}$ with ideal
point $v$.

Recall the setup before the analysis of situation 1. 
Let $\{ u_t \ | \ t \geq 0 \}$  be the collection
of unstable leaves intersected by the ray $l^1_{t_0}$.
The analysis is 
extremely similar to the analysis of situation 1, which shows all
cases produce continuity in the first step except
when $u_t$ converges to a leaf $u$ of $\wluf$ 
with ideal point $v$.
Then consider the stable leaves intersecting $u$.
The analysis of situation 1 shows continuity unless there 
is stable leaf
with ideal point $v$.
From now on the analysis is exactly the same as in situation
1, with unstable replaced by stable and vice versa.


%\vskip .2in
%Now look at the stable leaves intersected by $R_a$ which are the $S_t$.
%One of the ideal points of the $S_t$ is $p$. If the other one is
%$p$ we produce a non Hausdorff band. If the $s_t$ escape in $F$
%we obtain continuity on that side. Finally if the $s_t$ limit
%to a leaf we obtain a leaf $E_1$, non separated from $E_0$.
%Again look at $X_t$ $-$ the leaves intersecting $E_1$ 
%and $x_t = X_t \cap F$. Applying the same procedure
%we either end or continue this chain. Since there
%are non separated leaves in the chain, they are all periodic.
%After a while we produce a ${\bf Z} \oplus {\bf Z}$ in
%$\pi_1(M)$, contradiction. 
%
%What this shows is that the process needs to terminate after
%finitely many steps. 
%

\vskip .15in
So far we proved continuity of $\Psi$ at $v$
from the side of $e_0$ opposite to $l$.
The same works for the other side of $l$, producing
$l_0$ with similar properties as $e_0$.
We now must consider the regions between $e_0$ and
$e$, between $e$ and $l$ and between $l$ and $l_0$.

First consider the region between $e$ and $e_0$,
which occurs only when they are distinct. This implies
that the ray $e_0$ is a bounded distance from a
geodesic ray in $F$ with ideal point $v$.
Let $\{ \mu_t \ | \ t \geq 0 \}$ be a parametrization
of the stable leaves of $\wlsf$ through $e$.
Let $\mu^1_t$ be the component of $(\mu^1_t - e)$ 
in the side of $e$ we are considering. If some 
$\mu^1_t$ has ideal point $v$ then both ideal
points of $\mu_t$ are $v$ and $\mu_t$ is inside
a spike region. The same is true for $e$ and
so $e$ is a bounded distance from a geodesic
ray in $F$ with ideal point $a$.
%We refer to fig. \ref{vera} a.
Hence the region between $e$ and $e_0$ is a bounded
distance from a geodesic ray and we are finished in this
case.

%\blankfig{vera}{1.1}{a. The case $\mu^1_t$ has ideal point $v$,
%b. The remaining case.}

The remaining case to be analysed here is that $\mu^1_t$
has no ideal point $v$. Then $\mu^1_t$ does not escape
$F$ as $t \rightarrow \infty$, because $e_0$ is in that
side of $e$.
%, see fig. \ref{vera}, b.
So $\mu^1_t$ converges to a leaf $\mu$ which has ideal
point $v$.
Now consider a parametrization $\{ \nu_t \ | \ t \geq 0 \}$ of
the unstable leaves intersected by $\mu$.
Then $\nu_t$ converges to the leaf $e$.
If it converges to some other leaf, then $e$ is
a bounded distance from a geodesic ray in $F$ and we are done.
Otherwise it must be that some $\nu_t$ has ideal
point $v$. Therefore we exactly in the setup analysed
in situation 1 above. 

This shows continuity of $\Psi$ for the region between $e$ and
$e_0$ and similarly for the region between $l$ and $l_0$.

Finally we analyse the region $B$ between $e$ and $l$.
First notice there is no singularity in the interior
of $B$. Otherwise there would be a line leaf in
$B$ and hence a leaf with both endpoints $v$. 
It would have to be part of a spike region and the
spike region does not have any singularities in its 
interior.

Parametrize the leaves of $\wluf$ through $l$ as
$\{ e_s \ | \ s \geq 0 \}$ and similarly
those of $\wlsf$ through $e$ as $\{ l_t \ | \ t \geq 0 \}$.
Let $L, L_t$ leaves of $\wls$ with $l \subset L, \ l_t
\subset L_t$ and similarly define $E, E_t$.
There are 2 possibilities:





\vskip .1in
\noindent
1) Product case $-$ Any $l_t$ intersects every $e_s$ and
vice versa. 

Equivalently $\wlsf, \wluf$ define  a product structure 
in the region $B$ bounded by $l_0$ and $e_0$. If the $L_t$ escapes
in $\mi$ as $t \rightarrow \infty$, then there is a stable
product region defined by a segment in $L_0$. 
But then theorem \ref{prod} implies that
$\Phi$ is topologically conjugate to a suspension,
contradiction.
It follows that the 
$L_t$ converge to $H_1 \cup ... H_m$ as $t \rightarrow \infty$.
%As in case ??? for any $t$, $l_t$ converges to
Since the $l_t$ are stable leaves, it follows that
$F$ escapes up as $\Theta(F)$ appraches $\Theta(H_i)$.
This implies that $\Psi(e)$ limits to $(H_i)_+$ which
is then equal to $\Psi(v)$.
Similarly $E_s$ converges to $V_1 \cup ... V_n$ and
$F$ escapes down as $\Theta(F)$ approaches $\Theta(V_j)$. 
Hence 
$\Psi(l)$ limits to $(V_j)_- = \Psi(v)$.
If some $H_i$ intersects some
$V_j$, then 


$$(V_j \cap H_i)_+ \ = \ (H_i)_+ 
\ = \ \Psi(v) \ = \ 
(V_j)_- \ = \ (V_j \cap H_i)_-,$$

\noindent
contradiction.
%Let now $u_i$ in $F$ in the region $B$. 
Let now $\{ z_k \}$ be a sequence in $B$ converging
to $v$. The product structure implies that up to subsequence
we may assume that either $\ws(z_k)$ converges to
$H_i$ or $\wu(z_k)$ converges to $V_j$. 
This is analysed carefully in Claim 1 above, which
shows that $\Psi(z_k)$ must converge to $\Psi(v)$.
This shows continuity of $\Psi$ when restricted to the
region $B$.


%Suppose $u_i$ converges
%to $q$ not equal $\varphi(p)$. Then $u_i$ are in $\ws(p_{t_i})$ for some
%$t_i > 0$. Suppose first the $t_i$ does not converge to infinity.
%Up to subsequence we may assume that $t_i \rightarrow t_0$.
%\blankfig{regi}{1.1}{a. A region where there is a product structure.}
%Also $(v_i)_- = (q_{t_i})_-$. Here $t_i \rightarrow \infty$ else
%
%$$v_i \  \rightarrow  \ \ws(p_{t_0}) \cap \wu(q_{b_0}) \cap F.$$
%
%\noindent
%So $(v_i)_- = (q_{c_t})_-$ converges to $(V_j)_-$.
%If $v_i$ does not converge to $(V_j)_-$ then up to subsequence
%$\wwr(v_i)$ converges to $\wwr(v)$ and $v$ is in one of the $V_i$,
%say in $V_{i_0}$. But then $F$ escapes down so $v_i$ converges
%to $(V_j)_-$.
%Similarly  if $b_i$ does not converge to infinity. Finally if
%$t_i, b_i$ converge to infinity then
%
%$$(v_i)_+ = (p_{t_i})_+ \rightarrow (H_1)_+, \ \ 
%(v_i)_- = (q_{b_i})_+ \rightarrow (V_j)_-.$$
%
%\noindent
%Hence since the orbits are uniform quasigeodesics we obtain 
%$v_i \rightarrow (H_1)_-$ as required.
%This finishes the analysis in the product situation.
%

\vskip .1in
\noindent
2) Non product case.

There are $t, u > 0$ with $l_t \cap e_u = \emptyset$.
Consider one such $u$.
Let $a$ be the infimum of $t$ with $l_t \cap e_u = \emptyset$.
Now let $b$ be the infimum of $u$ with $l_a \cap e_u = \emptyset$.
Then $l_a \cap e_b = \emptyset$, but for any $0 \leq t < a$
and $0 \leq u < b$ one has 
$l_t \cap e_u \not = \emptyset$. 
Since $l_a \cap e_b = \emptyset$, then $L_a \cap E_b = \emptyset$.
It follows that $L_a, E_b$ form a perfect fit,
see fig. \ref{nene}, a.
If $\Theta(l_a)$ does not escape in $\Theta(L_a)$, then
there would be unstable boundary of $\Theta(F)$ in the
limit and that would keep $F$ from intersecting $E_b$,
contradiction. Hence $\Theta(l_a)$ escapes in $\Theta(L_a)$
and $\Theta(e_b)$ escapes in $\Theta(E_b)$.
Hence $\Psi(l_a)$ limits to $(L_a)_+$ and $\Psi(e_b)$ limits
to $(E_b)_-$. 
Also $l_a, e_b$ limit to $v$ in $\pin F$. 

Let $p_t = l_t \cap e$. If $\Theta(p_t)$ escapes in $\Theta(E)$,
then $\Psi(e)$ converges to $E_-$. Notice that $\Psi(e)$ converges
to $\Psi(v)$ so:


$$E_- \ = \ \Psi(v) \ = \ (L_a)_+ \ = \ (L_a \cap E)_+$$

\noindent
contradiction.
It follows that $\wwr(p_t)$ converges to
$\wwr(x)$ with $x$ in $E$.
Also $F$ has to escape up as $\Theta(F)$ approaches $\Theta(x)$
$-$ same as in Situation 1 above.
Hence $\Psi(e)$ limits to $x_+$. So

$$x_+ \ = \ \Psi(v) \ = \ (L_a)_+ \ = \ (p_a)_+$$

\noindent
Let $X = \wws(x)$.
Then $x, p_a$ are in 2 distinct orbits of $E$ with
the same positive ideal point.
Therefore theorem \ref{ident} implies that
$L_a,  X$ are connected by an even chain of lozenges
all intersecting $E$.
Let ${\cal C}$ be the first lozenge. It has a stable
side in $L_a$ and one unstable side, call it $D_1$ which
makes a perfect fit with $L_a$. Suppose first that $D_1$ is
in the component of $\mi - E$ opposite to $E_b$.
Then the other unstable side of ${\cal C}$, call it $D_2$
has to intersect $L_a$ in the other side of $E$.
%, see
%fig. \ref{nene}, a. 
Then $D_2$ must be some $E_t$,
let it be $E_{b'}$,
 see
fig. \ref{nene}, a. 
Then $\Theta(e_{b'})$ has to escape in $\Theta(E_{b'})$ 
or else one produces stable boundary to $\Theta(F)$ and
$\Theta(F)$ cannot limit to $\Theta(x)$ contradiction.
Hence $\Psi(e_{b'})$ converges to $\Psi(v)$ and 
also to $(E_{b'})_-$.
But then 

$$(E_{b'} \cap L_a)_- \ = \ 
(E_{b'})_- \ = \ \Psi(v) \ = \ (E_b)_- \ = \ (L_a)_+$$

\noindent
again a contradiction.


\begin{figure}
\centeredepsfbox{gm17.eps}
\caption{
a. Reaching before, b. Reaching at the exact time.}
\label{nene}
\end{figure}


This implies that $D_1$ is on the side of $E$ containing
$E_b$, see fig. \ref{nene}, b.

If there are only 2 lozenges in the chain from $L_1$ to
$X$, then  $D_1$ also makes a perfect fit with
$X$. Otherwise there are $D_2, ..., D_i$ all non separated
from $D_1$ and so that $D_i$ makes a perfect fit 
with $X$ and the $D_j$ are all in the boundary of the
chain of lozenges.
As seen in claim 1 above, $F$ cannot intersect
any $D_j$ ($1 \leq j \leq i$), but all
$\Theta(D_j)$ are contained in the unstable boundary
of $\Theta(F)$. Also $F$ escapes down as $\Theta(F)$
limits to $\Theta(D_j)$.
The set $\Theta(X)$ also has a line leaf which is
a stable boundary of $\Theta(F)$ and $F$ escapes
up when $\Theta(F)$ approaches $\Theta(X)$.

The same discussion applies to $L$, so there is 
$y$ in $L$, $Y = \widetilde W^u(y)$ with $\Theta(Y)$
having a line leaf in the unstable boundary
of $\Theta(F)$ and $F$ escapes down accordingly.
There are $C_1, ..., C_n$ leaves in $\wlu$, 
all non separated from each other and in the
boundary of the lozenges in the chain from $E_b$ to
$Y$ so that $C_1$ makes a perfect fit with $E_b$ and
$C_n$ makes a perfect fit with $Y$, see fig. \ref{nene}, b.
Finally $\Theta(C_j)$ has a line leaf in the
stable boundary of $\Theta(F)$ and $F$ escapes up
accordingly.

Let ${\cal E}$ be the region in $\mi$ bounded
by

$$E, \ L, \ X, \ Y, \ C_1, ..., C_n, \ D_1, ..., D_i$$

\noindent
Then ${\cal E} \cap F$ is exactly the region $B$
bounded by the rays $e$ and $l$.
Let $z_k$ in $B$ escaping to $v$.
Then the region ${\cal E}$ shows that
up to subsequence one of the following must occur:

\vskip .08in
1) $\ws(z_k)$ converges to either $X$ or $C_1$. 
The analysis of claim 1 above shows that
$\Psi(z_k)$ converges to either $X_+$ or $(C_1)_+$
both of which are equal to $\Psi(v)$.

2) $\wu(z_k)$ converges to either $Y$ or $D_1$.
Here $\Psi(z_k)$ converges to either $Y_-$ or
$(D_1)_-$ both of which are equal to $\Psi(v)$.

In any case this shows continuity of $\Psi$ in the
region $B$.
This finishes the non product case.

\vskip .05in
This finishes the proof of theorem \ref{exten},
the continuous extension theorem.
\end{proof}






%$(b_1)_+ = (L_{t_0} \cap E_0)_+$.
%As seen before this implies that $L_{t_0}$ and $A = \ws(b_1)$
%are connected by an even chain of lozenges, all
%intersecting a common stable???  leaf.


%If the asymptotic sides are in the side not containing $E_u, u > 0$,
%then there is an unstable side $E_{u_1}, u_1 < u_0$, see fig. \ref{nene}.
%But then
%
%$$(e_{u_1})_{\infty} \ = \ (E_{u_1})_- \ = \ (E_{u_1} \cap L_{t_0})_-$$
%
%\noindent
%But 
%
%$$(e_{u_0})_- \ = \ (E_{u_0})_- \ = \ (L_{t_0})_+,
%\ \ {\rm so} \ \ (L_{t_0})_+ \ = \ (E_{u_1} \cap L_{t_0})_-$$
%
%\noindent
%which is impossible.
%Therefore the asymptotic boundary lozenges are on the other side,
%see fig. \ref{nene}, b.


%So if $v_i$ is in $B$ and between $H_1, L_{t_0}$ and $E_0$ and
%it escapes $F$ then if it escapes $\oo$, it will
%converge to $(H_1)_-$. If it converges to a point
%in the boundary, then it will go to either $H_1$
%or to $V_1 \cup V_2 ...$. In either case either converges
%to $(H_1)_-$ or to $(V_i)_+$ which are equal. Hence we
%obtain continuity at $p$ in the other side of $l_{t_0}$.
%Similarly for the other side of $e_{u_0}$.
%Finally in the region between $l_{t_0}$ and $e_{u_0}$ if
%$\Theta(v_i)$ escapes, we have that 
%
%
%$$\wu(v_i) \rightarrow E_{u_0} \ \ {\rm and} \ \ 
%\ws(v_i) \rightarrow L_{t_0}. \ \ {\rm So} \ \ 
%(v_i)_- \rightarrow (L_{t_0})_+,
%(v_i))- \rightarrow (E_{u_0})_-$$
%
%\noindent
%which are equal. So again we obtain continuity as desired.
%This finishes the proof of the theorem.
%\end{proof}
%
%Conclusion: finite depth implies continuous extension.








\section{Foliations and Kleinian groups}

There are many similarities between foliations in hyperbolic
3-manifolds and Kleinian groups.
We refer to \cite{Mi,Can,Mar} for basic definitions concerning
degenerate and non degenerate Kleinian groups, in particular
singly and doubly degenerate groups.


If the foliation is $\rrrr$-covered then the limit set
of any leaf in $\mi$ is the whole sphere \cite{Fe3}. 
This corresponds to doubly degenerate surface
Kleinian groups \cite{Th1,Mi,Can,Mar,Bon}.
There is always a pseudo-Anosov flow which is transverse
to the foliation \cite{Fe7,Cal1}. If the flow is quasigeodesic then
the results of this article imply that the foliation
has the continuous extension property.

If the foliation has one sided branching, say branching down,
then limit sets of leaves can only have domain of discontinuity
``above" \cite{Fe3}. 
Let $F$ in $\fn$ and $\Lambda_F$ its limit set. If $p$ is not
in $\Lambda_F$, the $p$ is said to be {\em above} $F$ if
there is a neighborhood $V$ of $p$ in $\mi \cup \si$, so
that $V \cap \mi$ is on the positive side of $F$.
This corresponds to simply degenerate surface Kleinian
groups \cite{Th1,Mi,Can}. There are examples of foliations with one sided
branching transverse to 
suspension pseudo-Anosov flows
provided by Meigniez \cite{Me}. Suspension flows are always
quasigeodesic flows \cite{Ze}. 
The results of this article show the continuous extension property
for such foliations.
Under these conditions, the limit
sets are locally connected, the continuous extension provides
parametrizations of these limit sets.

Finally if there is branching in both directions, 
%for example
%in true Reebless finite depth foliations (not perturbations
%of fibrations). 
then there can be domain of discontinuity above
and below leaves. This corresponds to non degenerate
Kleinian groups \cite{Th1,Mi,Can}. These occur for example
in the case of finite depth foliations, where
the depth 0 leaves are not virtual fibers \cite{Fe6}.

There are many interesting questions:

\vskip .1in
\noindent
{\bf {Question 1}} $-$ Given a 
foliation $\fol$, is it  $\rrrr$-covered if and only if 
for every $F \in \fn$ then the limit set $\Gamma_F$ is $\si$?

The forward direction is true. The backwards direction
is true if there is a compact leaf \cite{Go-Sh}. In addition if there is one
leaf with limit set the whole sphere then all leaves have
limit set the whole sphere \cite{Fe3} $-$ whether $\fol$ is
$\rrrr$-covered or not.

\vskip .1in
\noindent
{\bf {Question 2}} $-$ Given $\fol$ an $\rrrr$-covered foliation, is there 
a quasigeodesic transverse pseudo-Anosov flow?

This is true in the case of slitherings or uniform foliations
as defined by Thurston \cite{Th5}. Examples are fibrations,
$\rrrr$-covered Anosov flows and many others.
There is always a transverse pseudo-Anosov flow, the
question is whether it is quasigeodesic.

\vskip .1in
\noindent
{\bf {Question 3}} $-$ Is there domain of discontinuity of $\Lambda_F$ only
above $F$ if and only if $\fol$ has one sided branching
in the negative direction?

This occurs for the examples constructed by Meigniez \cite{Me}.

\vskip .1in
\noindent
{\bf {Question 4}} $-$ Are the pseudo-Anosov flows constructed
by Calegari \cite{Cal2} and transverse to one sided branching foliations
quasigeodesic?


\vskip .1in
\noindent
{\bf {Question 5}} $-$ If $\fol$ has 2 sided branching is there
always domain of discontinuity above and below?
Is there a quasigeodesic pseudo-Anosov flow almost transverse to $\fol$?







{\footnotesize
{
\setlength{\baselineskip}{0.01cm}

\begin{thebibliography}{Fe-Mo}

{

%\setlength{\baselineskip}{0pt}
%\baselineskip0pt



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	surface bundles, I}, Proc. London Math. Soc. {\bf 82} (2001) 747-768.

\bibitem[Ro3]{Ro3} R. Roberts, {\em Taut foliations in punctured 
	surface bundles, II}, Proc. London Math. Soc. {\bf 83} (2001) 443-471.

\bibitem[Ro-St]{Ro-St} R. Roberts and M. Stein, {\em Group actions
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\bibitem[Ros]{Ros} Rosenberg, {\em Foliations by planes},
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\bibitem[Th1]{Th1} W. Thurston, {\em The geometry and topology of 3-manifolds},
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\bibitem[Th2]{Th2}  W. Thurston, {\em Three dimensional manifolds, Kleinian
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\bibitem[Th3]{Th3}  W. Thurston, {\em Hyperbolic structures on $3$-manifolds II:
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\bibitem[Th4]{Th4}  W. Thurston, {\em On the geometry and dynamics of
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\bibitem[Th5]{Th5}  W. Thurston, {\em 3-manifolds, foliations and circles I},
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\bibitem[Th6]{Th6}  W. Thurston, {\em 3-manifolds, foliations and circles II:
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\bibitem[Th7]{Th7}  W. Thurston, {\em The universal circle for foliations
	and transverse laminations}, lectures in M.S.R.I., 1997.

%\bibitem[Th7]{Th7}  W. Thurston, {\em The universal circle for foliations,
%Preprint}


%\bibitem[Wi]{Wi} R. Wilder, {\em Topology of Manifolds}, A.M.S. 
%Colloquium Publ. {\bf 32} (1949).

\bibitem[Ze]{Ze} A. Zeghib, {\em Sur les feuilletages g\'{e}od\'{e}siques
	continus des vari\'{e}t\'{e}s hyperboliques}, Inven. Math.
	{\bf 114} (1993) 193-206.



}
\end{thebibliography}



\noindent
Florida State University

\noindent
Tallahassee, FL 32306-4510, USA

}
}

\end{document}


math0502330.extracted_bib
\begin{thebibliography}{Fe-Mo}

{

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%\baselineskip0pt



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\bibitem[Ro2]{Ro2} R. Roberts, {\em Taut foliations in punctured 
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\bibitem[Ro3]{Ro3} R. Roberts, {\em Taut foliations in punctured 
	surface bundles, II}, Proc. London Math. Soc. {\bf 83} (2001) 443-471.

\bibitem[Ro-St]{Ro-St} R. Roberts and M. Stein, {\em Group actions
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\bibitem[Ros]{Ros} Rosenberg, {\em Foliations by planes},
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\bibitem[Th1]{Th1} W. Thurston, {\em The geometry and topology of 3-manifolds},
	Princeton University Lecture Notes, 1982.	

\bibitem[Th2]{Th2}  W. Thurston, {\em Three dimensional manifolds, Kleinian
	groups and hyperbolic geometry}, Bull. A.M.S. {\bf 6} (1982) 357-381.

\bibitem[Th3]{Th3}  W. Thurston, {\em Hyperbolic structures on $3$-manifolds II:
	Surface groups and $3$-manifolds that fiber over the circle}, preprint.

\bibitem[Th4]{Th4}  W. Thurston, {\em On the geometry and dynamics of
	diffeomorphisms of surfaces}, Bull. A.M.S. {19} (1988) 417-431.	


\bibitem[Th5]{Th5}  W. Thurston, {\em 3-manifolds, foliations and circles I},
	preprint.

\bibitem[Th6]{Th6}  W. Thurston, {\em 3-manifolds, foliations and circles II:
	Transverse asymptotic geometry of foliations}, preprint.

\bibitem[Th7]{Th7}  W. Thurston, {\em The universal circle for foliations
	and transverse laminations}, lectures in M.S.R.I., 1997.

%\bibitem[Th7]{Th7}  W. Thurston, {\em The universal circle for foliations,
%Preprint}


%\bibitem[Wi]{Wi} R. Wilder, {\em Topology of Manifolds}, A.M.S. 
%Colloquium Publ. {\bf 32} (1949).

\bibitem[Ze]{Ze} A. Zeghib, {\em Sur les feuilletages g\'{e}od\'{e}siques
	continus des vari\'{e}t\'{e}s hyperboliques}, Inven. Math.
	{\bf 114} (1993) 193-206.



}
\end{thebibliography}