Bounded geodesics in moduli space
Dmitry Kleinbock
In the moduli space of %Barak quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichm\"uller geodesic trajectories. The main tool is quantitative nondivergence of Teichm\"uller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.
math0304029.tex
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\title{Bounded geodesics in moduli space}
\author{Dmitry Kleinbock}
\address{Brandeis University, Waltham MA
02454-9110 {\tt kleinboc@brandeis.edu}}
\author{Barak Weiss}
\address{Ben Gurion University, Be'er Sheva, Israel 84105
{\tt barakw@math.bgu.ac.il}}

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\newtheorem*{main}{Theorem}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{step}{Step}
\newtheorem{claim}{Claim}
\newtheorem{remark}[thm]{Remark}
\newtheorem{conj}{Conjecture}
\newtheorem{question}{Question}

\begin{document}

%D
% Abstract is subject to change

   \begin{abstract}
In the moduli space of
%Barak
quadratic differentials over complex structures
on a surface,
we construct a set of full Hausdorff dimension of points
with bounded Teichm\"uller geodesic trajectories.
The main tool is quantitative nondivergence of
Teichm\"uller horocycles, due to Minsky and Weiss.
This has an application to billiards in rational polygons.

\end{abstract}

\maketitle

\section{Introduction}
Let $S$ be a compact orientable surface of genus $g$ with $n$
punctures,
%D with
where
$3g+n \geq 3$,
%
%{\comment  Dima: (why? same question as before, you want to
%avoid $g = 1$ which corresponds to square billiards and $G/\Gamma$,
%while  I'd rather had it in the picture as the ground case, why is it
%bad?)
%
%Barak: To convince you to omit the case $g=1$ I've inserted
%comments in the latex file, marked `genus', where we would have to say
%something extra for this case. }
and let
$\QQ$ be the moduli space of unit-area holomorphic quadratic
differentials over complex structures on $S$.
%genus - we will have to make sure that this is correctly formulated
%for g=1.
This is a noncompact orbifold, on which $G \df \SL(2,\R)$ acts
continuously
%B
with discrete stabilizers,
%{\comment  (maybe we should in addition remark here that a map
%$g\to gq$ is smooth and 1-1 in the neighborgood of identity for every $q$)},
preserving a smooth finite measure.
%D $\mu$.

For $t \in \R$, let $\displaystyle{ g_t = \left(\begin{array}{cc}
e^{t/2} & 0 \\ 0 & e^{-t/2} \end{array} \right) \in G \,, }$ and
denote the one-parameter subgroup $\{g_t : t \in \R\}$ by $F$. The
restriction of the $G$-action to $F$ defines a flow
on $\QQ$ called the {\em geodesic flow} (or sometimes the {\em
Teichm\"uller geodesic flow}). Its dynamical properties have been
extensively studied in
connection with various problems in ergodic theory and
geometry. See the surveys \cite{Howie -
handbook}, \cite{Mosher - cambridge} for more details.

It is known \cite{Masur - KC} that the
geodesic flow is ergodic
%{\comment  (I think it would be good to have a
%reference here -- I couldn't find this in \cite{Howie -
%handbook} and do not have \cite{Mosher - cambridge} handy; on the other
%hand, in \cite{Howie -
%handbook} they discuss measure supported on strata -- aren't strata
%ergodic components of your measure? I am somewhat confused)},
%D I think after a measure is mentioned we can just say ergodic
%D with respect to $\mu$
%D {\comment ($\mu$ was only introduced for the purpose of this sentence,
%D to clear up the confusion you pointed out. If there is a cleaner way
%D to clarify this, go ahead.)}
and hence a typical trajectory for the geodesic
flow is dense. It is natural (and of importance for applications) to
inquire as to the existence and abundance
%{\comment  (the term prevalence seems to be used mostly
%in the measure context, and does not seem to be appropriate here)}
of
atypical trajectories. Let us denote the
\hd\ of a subset $Y$ of a metric space $X$  by
$\dim(Y)$, and  say that $Y$ is {\it thick in} $X$
%Barak removed parantheses and stretched the italics
    if for any
nonempty open subset $W$ of $X$,
$
\text{dim}(W\cap
Y) = \text{dim}(W)
$
(i.e.~$Y$  has full \hd\ at any point of $X$).




In this note we discuss {\em bounded} trajectories, and prove the
following:


\begin{thm}
\label{thm: full dimension in space}
The set $\displaystyle{\{q\in \QQ : \text{the orbit }Fq\text{ is
bounded}\,\} }$ is thick in $\QQ$.
\end{thm}

Theorem \ref{thm: full dimension in space} follows from a more precise
result. Recall (see \S 2 for more details) that $\QQ$ is partitioned
into finitely many $G$-invariant suborbifolds called {\em strata}. Say
that $X \subset \QQ$ is {\em bounded in a stratum} if its closure is a
compact subset of a single stratum.

Let
$$F^+ \df \{g_t : t \geq 0\} \subset F\,.$$
For $s, \, \theta \in \R$ let
$$
h_s \df \left(\begin{array}{cc}
1 & s\\
0 & 1
\end{array}
\right), \ \ \ r_{\theta} \df
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}
\right)
\,,$$
%and $H\df\{h_s : s \in \R\}, \ R \df \{r_{\theta$.
and
$$
B\df\left\{\left(\begin{array}{cc}
a & 0\\
b & a^{-1}
\end{array}
\right):b\in\R,\ a\in\R\nz\right\}.$$



With this notation, we have:
\begin{thm}
\label{thm: more precise}
Let $q \in \QQ$. Then
\begin{itemize}
\item[(i)]
$\{s \in \R : F^+ h_sq \text{  is  bounded in  a
stratum}\, \}$ is thick in $\R$.
\item[(ii)]
For any
$f: \R \to B,$ \newline
$\{s \in \R : F^+ f(s) h_sq \text{  is  bounded in  a
stratum}\, \}$ is thick in $\R$.
\item[(iii)]
$\{\theta \in \R : F^+ r_{\theta} q \text{  is  bounded in  a
stratum}\, \}$ is thick in $\R$.
\item[(iv)]
%Barak changed g to x since g already used.
$\{x\in G : Fxq \text{  is  bounded in  a
stratum}\, \}$ is thick in $G$.
\end{itemize}
\end{thm}






%Barak
It follows from a result of Masur \cite{Masur -
log.laws} that the %BB
set in (iii) has measure zero, and it follows
easily that the same holds for the sets in (i), (ii), (iv).

\medskip

Note that Theorem \ref{thm: full dimension in space} follows
immediately from Theorem \ref{thm:
more precise}(iv) by considering the foliation of $\QQ$ into
$G$-orbits.
Theorem \ref{thm: more precise}(iii) has an interpretation in terms of
rational billiards,
%B: rewrote from here to end of paragraph.
which we now describe. Let $\mathcal{P}$ be a rational billiard table.
%D and let $\theta \in \R$.
%D
%D fixed \theta
Fix any $\theta
\in \R$, and for any $p \in \mathcal{P}$ consider a particle moving
at a constant velocity along a ray at angle $\theta$ starting at $p$. The
%D laws
law
of reflection (angle of
incidence equals angle of return)
%D give
gives rise to a dynamical system,
whose phase space $\mathcal{P}_{\theta}$ is
%D switched order
   the complement of a countable
set of lines in finitely many copies of $\mathcal{P}$.
See \cite{Howie - handbook} for more details.
Let $d$
denote the restriction of the Euclidean metric to
$\mathcal{P}_{\theta}$, and let
%D I think suppressing \theta is a bit confusing
%D $b_t:
$b^\theta_t:
   \mathcal{P}_{\theta} \to
\mathcal{P}_{\theta}$ denote the time $t$ map of this flow.
%D
%Barak
%We will omit the dependence of $b^\theta_t$ on $\theta$ whenever it does
%not cause confusion.

%Barak: I changed the next 2 paragraphs.


It can be shown (see e.g.\ \cite[\S 2]{Boshernitzan - rank two}) that
every billiard trajectory is {\em recurrent\/}, that is,
%D for
%D every $\theta$ and
  $\liminf_{t\to\infty}d(p,
b^\theta_tp) = 0$ for any  $p \in
\mathcal{P}_{\theta}$. Furthermore, one can use  \cite[Theorem
1.5]{Boshernitzan - recurrence} and the existence of one-dimensional
Poincar\'e sections for the flow
$(\mathcal{P}_{\theta},b^{\theta}_t)$ (see a discussion in \cite[\S
1.7]{Howie - handbook}) to derive a quantitative version  for generic
trajectories: namely,
%BB
given $\mathcal{P}$ and $\theta$, there is a constant $C$, such that
for almost every $p \in
\mathcal{P}_{\theta}$ one has
$$\liminf_{t\to\infty}t\cdot d(p, b^\theta_tp)
%D  < \infty
\le C
\,.$$

On the other hand it turns out that for a thick set of angles
$\theta$ it is not possible to replace $C$ in the right hand
side of the above inequality by an arbitrary small number:


\begin{cor}
\label{cor: billiards}

%\begin{itemize}
%\item[(i)]  The set
%$$\{\theta : \liminf_{t\to\infty}\,{{t}}\cdot
%d(p, b^\theta_tp)  = 0\quad \forall\,p \in
%\mathcal{P}_{\theta}\}$$
%has full measure.



%\item[(ii)]
The set
$$\{\theta : \exists\, c > 0 \text{ such that
}\liminf_{t\to\infty}\,{{t}}\cdot d(p, b^\theta_tp)  \ge c\quad
\forall\,p \in
\mathcal{P}_{\theta}\}$$ is thick in $\R$.
%\end{itemize}


\end{cor}


%\medskip

Note that if $\mathcal{P}$ is rectangular (or more generally
{\em integrable}, see \cite{Howie - handbook}), then $b^{\theta}_t:
\mathcal{P}_{\theta} \to \mathcal{P}_{\theta}$ is isomorphic to the
flow along a ray in the torus $\R^2/\Z^2$, and Corollary \ref{cor:
billiards} %is
%Barak
%assertions (i) and (ii) above are
is equivalent to the well-known fact (see \cite{Schmidt} and
the  references therein) that
%D
the set of
%D a thick set of $\theta \in \R$ are
{\em badly approximable\/} $\theta
\in \R$  (that is, those for which there exists
$c>0$ with $\displaystyle{|\theta - \frac{p}{q}| \geq
\frac{c}{q^2}}$ for all integers $p, \,q$)
%D form  a thick set.
is thick.
%D of measure zero.

\ignore{

\begin{cor}
\label{cor: billiards}

%\begin{itemize}
%\item[(i)]  The set
%$$\{\theta : \liminf_{t\to\infty}\,{{t}}\cdot
%d(p, b^\theta_tp)  = 0\quad \forall\,p \in
%\mathcal{P}_{\theta}\}$$
%has full measure.



%\item[(ii)]
The set
$$\{\theta : \exists\, c > 0 \text{ such that
}\liminf_{t\to\infty}\,{{t}}\cdot d(p, b^\theta_tp)  \ge c\quad
\forall\,p \in
\mathcal{P}_{\theta}\}$$ is thick in $\R$.
%\end{itemize}


\end{cor}


\medskip

Note that if $\mathcal{P}$ is rectangular (or more generally
{\em integrable}, see \cite{Howie - handbook}), then $b^{\theta}_t:
\mathcal{P}_{\theta} \to \mathcal{P}_{\theta}$ is isomorphic to the
flow along a ray in the torus $\R^2/\Z^2$, and Corollary \ref{cor:
billiards} is
%Barak
%assertions (i) and (ii) above are
is equivalent to the well-known fact (see \cite{Schmidt} and
the  references therein) that
%D a thick set of $\theta \in \R$ are
{\em badly approximable\/} $\theta
\in \R$  (that is, those for which there exists
$c>0$ with $\displaystyle{|\theta - \frac{p}{q}| \geq
\frac{c}{q^2}}$ for all integers $p, \,q$)
form  a thick set of measure zero.


It can be shown (see e.g. \cite[\S 2]{Boshernitzan - rank two}) that
every billiard trajectory is {\em recurrent\/}, that is, for
every $\theta$ and any  $p \in
\mathcal{P}_{\theta}$, $\liminf_{t\to\infty}d(p,
b^\theta_tp) = 0$. It is natural to inquire as to the rate of
recurrence, and our result places a lower bound on this rate for some
$(\theta, p)$. An upper bound can be obtained using a general argument of
Boshernitzan \cite{Boshernitzan - recurrence}, and the fact that a
Poincar\'e section in $\mathcal{P}_{\theta}$ for the flow
$b^{\theta}_t$ is one-dimensional. It follows from this argument that
for every rational billiard table $\mathcal{P}$, for every
$\theta \in \R$, for almost every $p \in \mathcal{P},$ one has
$$\liminf_{t\to\infty}t\cdot d(p, b^\theta_tp)  < \infty\,.$$
}


%\begin{question}
%Is it true that for every $\mathcal{P}, \, p \in P, \, \theta \in \R$
%one has
%$$\liminf_{t\to\infty}t\cdot d(p, b^\theta_tp)  < \infty\,?$$

%\end{question}

%For integrable $\mathcal{P}$ this follows immediately from
%Dirichlet's theorem.








%{\comment I decided against mentioning Dirichlet's theorem, both since
%it is not immediately related, and because I'm not sure I know how to
%prove it for a general rational billiard.}

\medskip

Our work is  part of a fruitful interaction between the study of
dynamics on quadratic differential spaces and dynamics on homogeneous
spaces of Lie groups, see the survey \cite{Cambridge} and the references
therein. Bounded trajectories on homogeneous spaces were extensively
studied due to their relation to linear forms which are badly
approximable by rationals, and results analogous to ours were obtained
by W.\ Schmidt \cite{Schmidt}, S.G.\ Dani \cite{Dani}, and, in most
general form, in
joint work of the first-named author and G.A.\ Margulis \cite{Dima -
Margulis}. See  \cite[\S 4.1b]{KSS survey} for references to other
related results.




Our construction of bounded trajectories is similar in some respects
to the one used in
\cite{Dima - Margulis}, but whereas in that paper a crucial estimate
was obtained by using the mixing property of the flow, in this paper
the estimate follows from quantitative nondivergence results for
horocycles obtained in \cite{with Yair}. This
%D new idea,
approach,
powered by nondivergence estimates for unipotent
trajectories
%D on
of homogeneous flows \cite{KM - manifolds}, can also be used to
reprove the abundance of bounded orbits for
%D the action of a
partially
hyperbolic diagonalizable
%D subgroup in
flows on
$\SL(n,\R)/\SL(n,\R)$,
and to obtain new applications to number theory.
%D
It was originally
introduced in the talk given by the first-named author  in July 2000 at a
Euroconference on  Ergodic Theory, Number Theory and
Geometric Rigidity  (Issac Newton Institute, Cambridge, UK).
%{\comment If you want to refer to talks, I think you already
%introduced it in NYU the preceding winter...}
%{\comment Dima: If we want to say more about
%these new results we can add another paragraph here. And we should
%also say something about the structure of the paper.
%
%Barak: I recommend saying nothing here. The new results will hopefully
%be written soon, stating them accurately requires too much space and
%if we want an informal statement, the above is enough; since the paper
%is so short there is no need to say anything about its structure.
%}



\medskip

{\bf Acknowledgements:}
We thank Lee Mosher for getting us
interested in this problem, and for encouraging remarks. We also thank
%BB
Howard Masur and Yair Minsky for teaching us about quadratic
differentials. This research was supported by BSF grant 2000247.

\medskip

\section{Terminology}
Let $\Mod(S)$ be the mapping class group, let $\til{\QQ}$ be the space
of unit-area holomorphic quadratic differentials over complex
structures on $S$, so that $\QQ = \QQQ/\Mod(S)$.
%genus - same remark as before.
Let $\pi: \til{\QQ}
\to \QQ$ be the natural quotient map. The space $\QQQ$ is a manifold
on which $G$ acts continuously, and $\pi$ is $G$-equivariant. Each
$\bq \in \QQQ$ can be defined as an equivalence
class of atlases of charts of the following type. Outside a
finite singularity set $\Sigma = \Sigma(\bq) \subset S$ we have charts
$S \sm \Sigma \to \C$ for which the transition functions $\C \to \C$
are of the form $z \mapsto \pm z +c$. The charts are required to
have $k$-pronged singularities around points of $\Sigma$, with $k \geq
3.$ At the punctures they are required to extend to the `filled-in'
surface, and the punctures may have $k$-prongs with $k \geq 1$.
The equivalence
relation is given by the natural action of
$\homeo_0(S)$ by pre-composition on each chart. An atlas is said to be
orientable if the transition functions can be taken to be of the form
$z \mapsto z +c$.
    See \cite[\S 4]{Howie -
handbook} or \cite[\S 4]{with Yair} for details.

The data consisting of the number of singularities of each type, and
the orientability of the atlas, assumes one of finitely many
possibilities. A level set for this data is called a {\em
stratum} in $\QQQ$. If $\mathcal{M} \subset \QQQ$ is a stratum we also
call its projection $\pi(\mathcal{M}) \subset \QQ$ a {\em stratum} in
$\QQ$. Any stratum in $\QQQ$ is a submanifold and any
stratum in $\QQ$ is a sub-orbifold. The strata
are invariant under the $\SL(2,\R)$-action.

A {\em   saddle connection\/}
%D with respect to
for
$\bq$ is a path
$\delta:(0,1)\to S \sm \Sigma$ whose image in any chart is a Euclidean
straight line and which extends continuously to a map from $[0,1]$ to the
      completion of $S$, mapping $\{0,1\}$ to singularities or punctures.
In all cases except $g=1, \, n=0$, there are saddle connections on the
surface. Since our results are immediate for this case, we assume from
now on that $(g,n) \neq (1,0)$.

     Integrating the local projections of
$d\delta$
%D we obtain
one obtains
a vector, denoted by
$\big(x(\delta,\bq),y(\delta,\bq)\big)$, well-defined up to a multiple of
$\pm 1$. Let $\LL_{\bq}$ denote the set of all saddle connections for
$\bq$. Since linear maps do not change the property of being a
straight line, we may identify $\LL_\bq$ with $\LL_{A\bq}$ for any
$A\in G$.  The action of
$G$ transforms the components of saddle connections linearly:
%Barak changed g to A since g already has a meaning.
\begin{equation}\label{eq: action on coordinates}
\text{for }A \in G\,, \ \left(
\begin{array}{c}
x(\delta,A\bq)\\
y(\delta,A\bq)
\end{array}
\right)
=\pm A
\left(
\begin{array}{c}
x(\delta,\bq)\\
y(\delta,\bq)
\end{array}
\right)\,.
\end{equation}

For a saddle connection $\delta\in\LL_\bq$, define its `length'
$l(\delta,\bq)$ by
\begin{equation*}
\label{eq: length def}
l(\delta,\bq)  \df \max \big(|x(\delta,\bq)|,|y(\delta,\bq)|\big)\,.
\end{equation*}

For $\vre>0$, let
$$K_{\vre} \df \pi \left(\{\bq \in \til{\QQ}:
\, \forall \delta \in \LL_{\bq}, \, l(\delta, \bq) \geq \vre\} \right).
$$
It is known that each $K_{\vre}$ is compact.
%genus: if g=1 then the K_{\vre} have to be redefined using closed
%loops instead of saddle connections, also there is no issue of strata.
Moreover, for each
stratum $\mathcal{M}$ in $\QQ$, each compact $K \subset \mathcal{M}$
is contained in $K_{\vre}$ for some $\vre$, and
$$\cl{\pi(\mathcal{M}) \cap K_{\vre}} \subset \pi(\mathcal{M}).$$
    Thus $X \subset \mathcal{M}$ is bounded in a stratum if and only if
$X \subset K_{\vre}$ for some $\vre>0$.

\section{From nondivergent horocycles to bounded geodesics}

Recall that the restriction of the $G$-action to the subgroup $\{h_s
:s \in \R\}$ defines the so-called {\em horocycle flow}. It was proved
in \cite{Veech}, using arguments of \cite{KMS}, that this flow does not
admit divergent trajectories. The main
ingredient for the proof of Theorem \ref{thm: more precise}
%Barak removed the (i) from the above ref
is a quantitative strengthening of this result, along the lines of
\cite{KM - manifolds}.
%{\comment Dima: (Here it seems
%appropriate
%to mention that the theorem below is analogous to and was motivated
%by a similar strengthening of a 1971 result of Margulis. Barak: Just
%added the last reference, don't want to spend too much space on the
%history.)}
   Here $|\cdot |$ stands for Lebesgue measure on $\R$.

%genus: the condition of this theorem is vacuous in case g=1 so the
%theorem has to be reformulated. Or else we can add the hypothesis
%that in this theorem g \geq 2.
\begin{thm}{\cite[Thm. 6.3]{with Yair}}
\label{thm: nondivergence}
There are positive constants $C, \, \alpha, \,
\rho_0$, depending only on $S$, such that if $\bq \in \QQQ$, an
interval $I \subset \R$, and $0<\rho \leq \rho_0$ satisfy:
\begin{equation}
\label{eq-condition}
{\rm for \ any \ } \delta \in \LL_{\bq}, \ \sup_{s \in I} l(h_s\bq, \delta)
\geq \rho,
\end{equation}
then for any $0<\vre<\rho$ we have:
\begin{equation}
\label{eq - precise yet long}
|\{s \in I: h_s \pi(\bq) \notin K_{\vre} \}| \leq
C\left(\frac{\vre}{\rho}\right)^{\alpha}|I|.
\end{equation}

\end{thm}

\medskip

{\bf Proof of Theorem \ref{thm: more precise}.}
%D
We start with part (i).
Let $J \subset \R$ be an interval and let $q_0 = \pi(\bq_0) \in \QQ$. We
need to prove that the Hausdorff dimension of the set
\begin{equation}
      \label{eq: set}
\{ s \in J : \, F_+ h_{s}q_0 \mathrm{\ is \
bounded \ in \ a \ stratum} \,\}
\end{equation}
is equal to $1$.
%D By rescaling, without loss of generality
%D we can assume that $|J| = 1$.
Denote the length of $J$ by $r$, and assume, as we may without loss of
generality, that $J = [0,r]$ and $r\le 1$.



Let $\eta>0$ be arbitrary, and
let $C, \, \alpha, \, \rho_0$ be as in Theorem \ref{thm:
nondivergence}. Choose $\vre>0$ so that
%D
$q_0\in K_{\vre}$, and also
$$C \left (\frac{\vre}{ \rho_0}\right) ^{\alpha} < \eta\,,$$
and let
\[
K' \df K_{\vre}, \ \ \ \ \ K''
\df
\bigcup_{0\le s \le r
} h_s K'\,.
\]

Then there is $\vre_0$, depending only on $\vre$,
such that $K'' \subset K_{\vre_0}$. Hence for
each $q \in K''$, each $\bq \in \pi^{-1}(q),$ and
each $\delta  \in
\LL_{\bq}$, we have
\begin{equation}
\label{eq: vre_0}
l(\delta , \bq) = \max \big(|x(\delta,\bq)|,|y(\delta,\bq)|\big) \geq
\vre_0\,.
\end{equation}



\begin{claim}
\label{claim: achieve large}
There exists $t_1>0,$ depending only on
$\vre_0$ and $r$,
     such that for
every
$q \in K'', \, \bq \in \pi^{-1}(q)$ and $\delta \in
\LL_{\bq}$,
\begin{equation}
\label{eq: ii}
     \sup_{0\le s \le r}
|x(\delta,
g_{t_1} h_{s}  \bq)|
\geq \rho_0\,.
\end{equation}

      \end{claim}

%{\comment  Dima: (I found it personally very irritating when first
%read your draft that proofs of claims are not marked as such, making
%it hard to understand whether you follow the claim by its proof or
%just assume it as is. This is why I inserted `Proof's and $\qed$s.)
%
%Barak: I erased them because I'm used to the convention that $\qed$
%marks the end of the proof. That's why I indicated with words and a
%``medskip'' where the proofs of the claims end. I prefer to leave it
%as it is.
%}

%D
To prove the claim,
let us first show that
\begin{equation}
\label{eq: i}
%D   \sup_{0\le s \le 1}
%D   |x(\delta, h_{s} \bq)| \geq \vre_0/2\,.
\max\big(|x(\delta, \bq)|,|x(\delta, h_{r} \bq)|\big) \geq
\vre_0r/2\,.
\end{equation}
Let
$x=x(\delta,  \bq)$, $y=y(\delta,
     \bq)$. If $|x| \geq
\vre_0r/2$, there is nothing to prove. So by (\ref{eq: vre_0}) we may
assume that $|y| \geq \vre_0$. But then
\[
     |x(\delta, h_{r} \bq)|
\stackrel{\equ{eq: action on
coordinates}}{=} |x   + ry| \ge \vre_0r -
\vre_0r/2=\vre_0r/2 \,.
\]
Setting $t_1 \df 2\log \frac{2\rho_0}{\vre_0r}$, one obtains
(\ref{eq: ii}) from (\ref{eq: i}) via (\ref{eq: action on
coordinates}). This proves the claim.

\medskip

Enlarging $t_1$ if necessary, assume that
$$N \df e^{t_1} \in \N\,.$$

\begin{claim}
\label{claim: return}
Suppose $q' \in \QQ
%Barak.
%\pi(\mathcal{M})
$ and $t_0 \geq 0$ are
such that
$g_{t_0} h_s q' \subset K''$ for any $\,0 \le s \le re^{-t_0}$.
Subdivide
$[0,re^{-t_0}]$
into $N$
subintervals $J_1, \ldots, J_N$ of length $re^{-t_0}/N=re^{-(t_0+t_1)}$,
and let
$$\mathcal{I} \df \left \{i \in \{1, \ldots, N\} :
g_{t_1+t_0}h_{s}q' \in K'' \ \forall\,s\in J_i\right \}
.$$
Then $\# \mathcal{I} \geq (1 -\eta)N.$
\end{claim}

%D
In order to prove this claim,
we apply Theorem \ref{thm: nondivergence} with
$I = [0,re^{t_1}]$ and
$q =  g_{t_1+t_0}q'$, that is, to the horocycle
\[
\{h_s g_{t_1+t_0}q' : 0 \le s \le re^{t_1}\} = \{g_{t_1}h_s g_{t_0}q':
0 \le s
\le r\}\,.
\]
     Assumption (\ref{eq-condition}), with
$\rho=\rho_0$, is valid by (\ref{eq: ii}).

We obtain:
\begin{equation}
      \label{eq: intersect most of the time}
\begin{split}
|\{0 \le s \le re^{-t_0}:   g_{t_1+t_0}h_sq' \notin K' \}| &=
e^{-(t_1 + t_0)}|\{0 \le s \le re^{t_1}:  h_s g_{t_1+t_0}q' \notin K'
\}|\\
     &\leq
e^{-(t_1 + t_0)} C\left(\frac{\vre}{\rho_0}\right)^{\alpha}re^{t_1} < \eta
re^{-t_0}\,.
\end{split}
\end{equation}

Now let
$$\mathcal{I}_0 \df \{ i \in \{1, \ldots, N\} : \,
g_{t_1+t_0} h_{s}q' \in K' \text{ for some }s\in J_i \}\,.
$$
Then (\ref{eq: intersect most of the time}) implies that
$\# \mathcal{I}_0 \geq (1-\eta)N.$
A simple calculation
using
the definition of $K''$ shows that
$\mathcal{I}_0 \subset \mathcal{I}$, and the claim follows.

\medskip

We now construct a subset $D \subset J$, of large Hausdorff dimension,
by iterating Claim \ref{claim:
return}. At stage $m$ of the construction we have a subdivision of $J$
into $N^m$ subintervals of length $N^{-m}$, and a collection of
$[(1-\eta)N]^m$ of these subintervals with the property that if $I$ is
one of these intervals then
$$ g_{kt_1}h_s q_0 \subset  K'' \text{ for any }s\in I\text{ and }\ k=0,
\ldots, m\,.$$ For each subinterval $I$ belonging to this collection, we
apply Claim
\ref{claim: return} to $I$ shifted by its left endpoint, with $t_0=mt_1$
and an appropriate
$q' \in \pi(\mathcal{M})$, and obtain that there are at least $(1-\eta)N$
subintervals of $I$ which satisfy the conclusion of the claim. The union
of these subintervals over all subintervals $I$ comprises the collection
for the \linebreak
$(m+1)$-st step.

Let $D$
be the set of intersection points of all
sequences of nested subintervals in the above construction.




A well-known result
     (see  \cite[Proposition 2.2]{McMullen})
implies that the Hausdorff dimension of $D$ is at
least
$$1-\frac{\log(1-\eta)}{-\log(N)} \geq 1+\log(1-\eta)\,.$$
If $s \in
D$, then
%for any $k \in \N, \ \ g_{kt_1}h_{s}q \in K''.$
$g_{kt_1}h_{s}q \in K''$ for any $k \in \Z_+$.
This
implies that
$$F_+ h_{s} q_0 \subset K \df \bigcup_{0\le t \le t_1
} g_t K''\,,$$
where $K$ is compact.

Since $\eta$ was chosen arbitrarily and since $1+ \log(1-\eta)$ tends
to $1$ as $\eta\to 0$, we obtain that the set \equ{eq: set} has Hausdorff
dimension $1$, as required.

\medskip

For part (ii), note that for every $b \in B$, the set
%Barak changed this part.
$$\{g_tbg_{-t} : t \geq 0\} \subset G$$
is precompact, hence
$$F^+ f(s) h_sq = \big\{\big(g_t f(s)
g_{-t}\big) g_t  h_sq : t\ge 0\big\}$$ is bounded if and only if so is
$F^+ h_sq$; thus (ii) follows from (i). The claim (iii) is a special
case of (ii): it is easy to represent $r_\theta$ in the form $r_\theta =
f(\theta) h_\theta$ where $f(\theta) \in B$.

Finally, to prove (iv), we
%D
follow the argument of \cite[\S 1.5]{Dima -
Margulis}. Take
$U\subset G$  of the form
$V^- V V^0$, where $V^-$, $V$  and $V^0$ are
neighborhoods of identity in
$$
H^-\df \left\{\left(\begin{array}{cc}
1 & 0\\
s & 1
\end{array}
\right) : s\in \R \right\}
, \ \
%Barak
H \df \{h_s: s \in \R\} \ \ \mathrm{and \ } F \
\mathrm{respectively},$$ such
that the multiplication map $V^-\times V\times V^0\to U$
%,$ $(h^-,h,
%h_0)\mapsto h^-hh^0$,
is bi-Lipschitz. It is enough to show that for any $q$,
\begin{equation}
      \label{eq: dim 3}
\dim\big(\{h \in U : Fhq\text{ is
bounded}\,\}\big) = 3\,.
\end{equation}
Let $C$ be the set $\{h\in U : F^+hq\text{ is
bounded}\,\}$. From part (i) it
follows that for any $h^0\in V^0$, $\dim(C\cap Vh^0) = 1$. In view of the
slicing properties of the \hd\ (see \cite{Falconer} or \cite[Lemma
1.4]{Dima - Margulis}),  one deduces that
\begin{equation}
      \label{eq: dim 2}
\dim(C\cap V V^0) = 2\,.
\end{equation}


Now for any $h\in C$ choose a neighborhood $V^-(h)$
of identity in $H^-$ such that
$V^-(h)h\subset U$. Then, by the argument used in the proof of (ii), one
necessarily has
$V^-(h)h\subset C$. Applying part (i) of the theorem
with
$$F^-\df \{g_t : t\le 0\}$$ in place of $F^+$ and $H^-$ in place
of $H$ (this can be justified e.g.\ by considering an automorphism
$q\mapsto \left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}
\right)q$ of $\QQ$, which leaves $G$-orbits and lengths of saddle
connections invariant, and sends the action of $H$ and $F^+$ to that of
$H^-$ and $F^-$ respectively),
one concludes that
$$
\forall\,h\in C\cap V V^0\quad\dim\big(\{h^- \in V^- : Fh^-
hx\text{ is bounded}\,\}\big) = 1\,,
$$
which, together with \equ{eq: dim 2}, yields \equ{eq: dim 3}.
\qed

%\medskip
%
%{\bf Proof of Theorem \ref{thm: more precise}(ii-iv).}
%
%{\comment  (I'll write it later, this is not a big deal.)}
%\qed


\section{Application to rational billiards}



%{\comment Perhaps the first paragraph here belongs in section 2? I
%have no strong opinion about this.}

Recall that any $\bq \in \QQQ$ defines a pair of transverse measured
foliations on $S \sm \Sigma(\bq)$, called the {\em horizontal\/} and
{\em vertical\/} foliations respectively, and that $g_t$ acts by
contracting (resp.\ expanding) the measure transverse to horizontal
(resp.\ vertical) leaves by a factor of $e^{t/2}$. The Euclidean metric
on
%D
   a billiard table
$\mathcal{P}$ determines a flat Riemannian metric on
$S
\sm
\Sigma (\bq)$. The length of $\delta \in \LL_{\bq}$ with respect to this
metric is at least as large as $l(\delta, \bq)$. Recall also that any
line segment is a length-minimizing path in its homotopy class with
endpoints fixed, and any free homotopy class has a length-minimizing
representative which is a concatenation of saddle connections.

Given $\mathcal{P}$, there is a standard construction (see
\cite{KMS} or \cite{Howie - handbook}) of a
%Barak
closed surface $S$ of genus $g \geq
1$, with a  quadratic differential $\bq$, such
that, for each $\theta \in
\R$, the  flow $b^{\theta}_t : \mathcal{P}_{\theta} \to
\mathcal{P}_{\theta}$ is isomorphic to the unit-speed flow in $S \sm
\Sigma(\bq)$ along vertical leaves for $r_{\theta} \bq$.
%Barak
The case $g=1$ occurs if and only if $\mathcal{P}$ is integrable, and
in this case Corollary \ref{cor: billiards} follows
from the fact that badly approximable numbers form a thick subset of
the reals. So assume $g \geq 2$.

%The main ingredient of the proof of Corollary \ref{cor:
%billiards} is the following correspondence between the behavior
%of billiard orbits and Teichm\"uller geodesics, which seems to be
%of independent interest:



%\begin{lem}
%\label{lem: correspondence}
%Let  $\mathcal{P}$ and $\bq$ be as above. Then:
%\begin{itemize}
%\item[(i)] if
%$F^+ r_{\theta} \pi(\bq)$  is  not bounded in a stratum, then
%$$\liminf_{t\to\infty}\,{{t}}\cdot d(p, b^\theta_tp)  = 0\quad\forall\,p
%\in
%\mathcal{P}_{\theta}\,;$$
%\item[(ii)]
% if
%$F^+ r_{\theta} \pi(\bq)$  is  bounded in a stratum, then for some $c >
%0$ one has $$\liminf_{t\to\infty}\,{{t}}\cdot d(p, b^\theta_tp)  \ge
%c\quad
%\forall\,p \in
%\mathcal{P}_{\theta}
%\,.$$
%\end{itemize}
%\end{lem}

\medskip

{\bf Proof of Corollary \ref{cor: billiards}.} Suppose $\theta$
belongs to the thick subset of $\R$ for which $F^+ r_{\theta}
\pi(\bq)$ is bounded in a stratum. Then there exists
$0<\vre<1$ with
\begin{equation}
\label{eq: fix the compact}
F^+r_{\theta}\pi(\bq) \subset K_{\vre}\,.
\end{equation}
Let $c \df {{\vre^2}/{2}<1}$; we claim that
${d(b^\theta_tp,p)\ge{c}/{t}}$ for any $p
\in
\mathcal{P}_{\theta}$  and $t\geq 1$.
Indeed, assuming  that there are $p \in \mathcal{P}_{\theta}$ and $t\geq
1$ with
${d(b^\theta_tp,p) < {c}/{t}}$, let
$$t_0 \df 2 \log t - \log c \geq 0\,.$$


With respect to $r_{\theta} \bq$, let $\mathcal{C}_1$ be the vertical
segment from $p$ to $b^\theta_tp$, let $\mathcal{C}_2$ be a
length-minimizing path from $b^\theta_tp$ to $p$, and let $\mathcal{C}$
be the concatenation of
$\mathcal{C}_1$ and $\mathcal{C}_2$. The length of $\mathcal{C}_1$
with respect to $r_{\theta} \bq$ is $t \geq 1$ and the length of
$\mathcal{C}_2$ is less
than $c/t<1$. Since $\mathcal{C}_1$ is length-minimizing in its
homotopy class with fixed endpoints, $\mathcal{C}$ is homotopically
nontrivial.

With respect to $g_{t_0} r_{\theta} \bq$, the length of
$\mathcal{C}_1$ is $te^{-t_0/2} = \sqrt{c}$, and the length of
$\mathcal{C}_2$ is less than
$e^{t_0/2}c/t=\sqrt{c}$. Thus the length of $\mathcal{C}$ with respect
to $g_{t_0}r_{\theta} \bq$ is less than $\vre$. Since
$\mathcal{C}$ is not homotopically trivial, the shortest
representative in its free homotopy class is a nontrivial
concatenation of saddle
connections. In particular there is $\delta \in
\LL_{\bq}$ with $l(\delta, g_{t_0}r_{\theta} \bq) < \vre$, which
contradicts (\ref{eq: fix the compact}). \qed

%\medskip

%{\bf Proof of Corollary \ref{cor: billiards}.}  Part (i) follows from
%Lemma \ref{lem: correspondence}(i) and the result of \cite{Masur -
%log.laws} mentioned in the introduction, while (ii) is a consequence
%of Lemma \ref{lem: correspondence}(ii) and Theorem \ref{thm: more
%precise}(iii).
%\qed



%\medskip
%{\comment (If the proof of (i) goes well, I think it makes sense to
%remark that, unexpectedly, recurrence properties in a fixed direction
%are independent of a starting point. And also about the analogy to
%dioph.approximation and hom.spaces.)}


\begin{thebibliography}{99}
\bibitem[Bo1]{Boshernitzan - rank two} M.\ Boshernitzan, {\em Rank two
interval exchange transformations}, Erg. Th. Dyn. Sys. {\bf 8} (1988)
379--394.


\bibitem[Bo2]{Boshernitzan - recurrence} M.\ Boshernitzan, {\em
Quantitative recurrence results},  Invent. Math. {\bf 113} (1993),
617--631.


\bibitem[Da]{Dani} S.G.\ Dani, {\em Bounded orbits of flows on
homogeneous spaces}, Comm.\ Math.\ Helv.\ {\bf 61} (1986), 636--660.


\bibitem[Fa]{Falconer} K. Falconer, {\bf Fractal geometry}, Wiley,
Chichester, 1990.

\bibitem[KMS]{KMS} S.\ Kerckhoff, H.\ Masur and J.\ Smillie, {\em
Ergodicity of billiard flows and quadratic differentials},
Ann.\ Math.\ {\bf 124} (1980), 293--311.

\bibitem[KM1]{Dima - Margulis} D.\  Kleinbock and G.A.\ Margulis,
{\em Bounded orbits of nonquasiunipotent flows on homogeneous spaces},
Amer.\ Math.\ Soc.\ Translations {\bf 171} (1996), 141--172.


\bibitem[KM2]{KM - manifolds}  D.\ Kleinbock and G.A.\
Margulis, {\em Flows on homogeneous spaces and Diophantine
approximation on manifolds,} Ann.\ Math.\ {\bf 148} (1998), 339--360.



\bibitem[KSS]{KSS survey}D.\ Kleinbock, N.\ Shah and A.\ Starkov,
{\em Dynamics of subgroup actions on homogeneous spaces of Lie groups
and  applications to number theory},   in:  {\bf Handbook on Dynamical
Systems}, Volume 1A, Elsevier Science,  North Holland, 2002, pp.\
813--930.

\bibitem[M1]{Masur - KC} H.\ Masur, {\em Interval Exchange
Transformations and Measured Foliations}, Ann. Math. {\bf 115} (1982)
169--200.


\bibitem[M2]{Masur - log.laws} H.\ Masur, {\em Logarithmic law for
geodesics in moduli space}, in: {\bf Mapping class groups and moduli
spaces of Riemann surfaces},
229--245,  Contemp.\ Math.\ {\bf 150},  Amer.\ Math.\ Soc., Providence,
RI, 1993.

\bibitem[MT]{Howie - handbook} H.\ Masur and S.\ Tabachnikov, {\em
Rational billiards and flat structures}, in:  {\bf Handbook on Dynamical
Systems}, Volume 1A, Elsevier Science,  North Holland, 2002, pp.\
1015--1089.

%\bibitem[Ma]{Masur - ICM} H. Masur, {\em Teichm\"uller space,
%dynamics, probability}, in {\bf Proc. Intl. Con. Math. 1994} 836--849,
%Birkh\"auser.

\bibitem[Mc]{McMullen}  C. McMullen, {\em  Area and \hd\ of Julia sets
of entire functions},  Trans.\ Amer.\ Math.\ Soc.\ {\bf 300}  (1987),
329--342.


\bibitem[Mo]{Mosher - cambridge} L.\ Mosher, {\em Schottky subgroups of
mapping class groups and the geometry of surface-by-free groups}, in:
{\bf Rigidity in Dynamics and Geometry}, M.\ Burger and A.\ Iozzi, eds.,
Springer, 2002, pp.\ 309--319.





\bibitem[MW]{with Yair} Y.N.\ Minsky and B.\ Weiss, {\em
Nondivergence of horocyclic flows on moduli space} J.\ reine
angew.\ math.\ {\bf 552} (2002), 131--177.

\bibitem[Sc]{Schmidt} W.M.\ Schmidt, {\em On badly approximable
numbers and certain games}, Trans.\ Amer.\ Math.\ Soc.\ {\bf 110} (1964),
493--518.

%\bibitem[Ur]{Urbanski} M.\ Urbanski, {\em The Hausdorff dimension of
%the set of points with non-dense orbit under a hyperbolic dynamical
%system,} Nonlinearity {\bf 4} (1991), 385--397.

\bibitem[Ve]{Veech} W.A.\ Veech, {\em Measures supported on the
set of uniquely ergodic directions of an arbitrary holomorphic
1-form}, Erg. Th. Dyn. Sys. {\bf 19} (1999), 1093--1109.

\bibitem[We]{Cambridge} B.\ Weiss, {\em Dynamics on parameter spaces:
submanifold and fractal subset questions},  in:
{\bf Rigidity in Dynamics and Geometry}, M.\ Burger and A.\ Iozzi, eds.,
    Springer, 2002, pp.\ 425--439.

\end{thebibliography}
\end{document}


Using Theorem \ref{thm: more precise}(iii) suppose
$\theta$ is in the
thick subset of $\R$ for which there is $0<\vre<1$ with
\begin{equation}
\label{eq: fix the compact}
F^+r_{\theta}\pi(\bq) \subset K_{\vre}.
\end{equation}

Let $c \df \displaystyle{\frac{\vre^2}{2}<1}.$ Suppose if possible that
there are $p \in \mathcal{P}$ and $t\geq 1$ with
$\displaystyle{d(b_tp,p)<\frac{c}{t}}.$ Let
$$t_0 \df 2 \log t - \log c \geq 0.$$


With respect to $r_{\theta} \bq$, let $\mathcal{C}_1$ be the vertical
segment from $p$ to $b_tp$, let $\mathcal{C}_2$ be a length-minimizing
path from $b_tp$ to $p$, and let $\mathcal{C}$ be the concatenation of
$\mathcal{C}_1$ and $\mathcal{C}_2$. The length of $\mathcal{C}_1$
with respect to $r_{\theta} \bq$ is $t \geq 1$ and the length of
$\mathcal{C}_2$ is less
than $c/t<1$. Since $\mathcal{C}_1$ is length-minimizing in its
homotopy class with fixed endpoints, $\mathcal{C}$ is homotopically
nontrivial.

With respect to $g_{t_0} r_{\theta} \bq$, the length of
$\mathcal{C}_1$ is $te^{-t_0/2} = \sqrt{c}$, and the length of
$\mathcal{C}_2$ is less than
$e^{t_0/2}c/t=\sqrt{c}$. Thus the length of $\mathcal{C}$ with respect
to $g_{t_0}r_{\theta} \bq$ is less than $2 \sqrt{c} \leq \vre$. Since
$\mathcal{C}$ is not homotopically trivial, the shortest
representative in its free homotopy class is a nontrivial
concatenation of saddle
connections. In particular there is $\delta \in
\LL_{\bq}$ with $l(\delta, g_{t_0}r_{\theta} \bq) < \vre$, which
contradicts (\ref{eq: fix the compact}).
math0304029.extracted_bib
\begin{thebibliography}{99}
\bibitem[Bo1]{Boshernitzan - rank two} M.\ Boshernitzan, {\em Rank two
interval exchange transformations}, Erg. Th. Dyn. Sys. {\bf 8} (1988)
379--394.


\bibitem[Bo2]{Boshernitzan - recurrence} M.\ Boshernitzan, {\em
Quantitative recurrence results},  Invent. Math. {\bf 113} (1993),
617--631.


\bibitem[Da]{Dani} S.G.\ Dani, {\em Bounded orbits of flows on
homogeneous spaces}, Comm.\ Math.\ Helv.\ {\bf 61} (1986), 636--660.


\bibitem[Fa]{Falconer} K. Falconer, {\bf Fractal geometry}, Wiley,
Chichester, 1990.

\bibitem[KMS]{KMS} S.\ Kerckhoff, H.\ Masur and J.\ Smillie, {\em
Ergodicity of billiard flows and quadratic differentials},
Ann.\ Math.\ {\bf 124} (1980), 293--311.

\bibitem[KM1]{Dima - Margulis} D.\  Kleinbock and G.A.\ Margulis,
{\em Bounded orbits of nonquasiunipotent flows on homogeneous spaces},
Amer.\ Math.\ Soc.\ Translations {\bf 171} (1996), 141--172.


\bibitem[KM2]{KM - manifolds}  D.\ Kleinbock and G.A.\
Margulis, {\em Flows on homogeneous spaces and Diophantine
approximation on manifolds,} Ann.\ Math.\ {\bf 148} (1998), 339--360.



\bibitem[KSS]{KSS survey}D.\ Kleinbock, N.\ Shah and A.\ Starkov,
{\em Dynamics of subgroup actions on homogeneous spaces of Lie groups
and  applications to number theory},   in:  {\bf Handbook on Dynamical
Systems}, Volume 1A, Elsevier Science,  North Holland, 2002, pp.\
813--930.

\bibitem[M1]{Masur - KC} H.\ Masur, {\em Interval Exchange
Transformations and Measured Foliations}, Ann. Math. {\bf 115} (1982)
169--200.


\bibitem[M2]{Masur - log.laws} H.\ Masur, {\em Logarithmic law for
geodesics in moduli space}, in: {\bf Mapping class groups and moduli
spaces of Riemann surfaces},
229--245,  Contemp.\ Math.\ {\bf 150},  Amer.\ Math.\ Soc., Providence,
RI, 1993.

\bibitem[MT]{Howie - handbook} H.\ Masur and S.\ Tabachnikov, {\em
Rational billiards and flat structures}, in:  {\bf Handbook on Dynamical
Systems}, Volume 1A, Elsevier Science,  North Holland, 2002, pp.\
1015--1089.

%\bibitem[Ma]{Masur - ICM} H. Masur, {\em Teichm\"uller space,
%dynamics, probability}, in {\bf Proc. Intl. Con. Math. 1994} 836--849,
%Birkh\"auser.

\bibitem[Mc]{McMullen}  C. McMullen, {\em  Area and \hd\ of Julia sets
of entire functions},  Trans.\ Amer.\ Math.\ Soc.\ {\bf 300}  (1987),
329--342.


\bibitem[Mo]{Mosher - cambridge} L.\ Mosher, {\em Schottky subgroups of
mapping class groups and the geometry of surface-by-free groups}, in:
{\bf Rigidity in Dynamics and Geometry}, M.\ Burger and A.\ Iozzi, eds.,
Springer, 2002, pp.\ 309--319.





\bibitem[MW]{with Yair} Y.N.\ Minsky and B.\ Weiss, {\em
Nondivergence of horocyclic flows on moduli space} J.\ reine
angew.\ math.\ {\bf 552} (2002), 131--177.

\bibitem[Sc]{Schmidt} W.M.\ Schmidt, {\em On badly approximable
numbers and certain games}, Trans.\ Amer.\ Math.\ Soc.\ {\bf 110} (1964),
493--518.

%\bibitem[Ur]{Urbanski} M.\ Urbanski, {\em The Hausdorff dimension of
%the set of points with non-dense orbit under a hyperbolic dynamical
%system,} Nonlinearity {\bf 4} (1991), 385--397.

\bibitem[Ve]{Veech} W.A.\ Veech, {\em Measures supported on the
set of uniquely ergodic directions of an arbitrary holomorphic
1-form}, Erg. Th. Dyn. Sys. {\bf 19} (1999), 1093--1109.

\bibitem[We]{Cambridge} B.\ Weiss, {\em Dynamics on parameter spaces:
submanifold and fractal subset questions},  in:
{\bf Rigidity in Dynamics and Geometry}, M.\ Burger and A.\ Iozzi, eds.,
    Springer, 2002, pp.\ 425--439.

\end{thebibliography}