Topological entropy and Burau representation
Boris Kolev
Let $f$ be an orientation-preserving homeomorphism of the disk $D^{2}$, $P$ a finite invariant subset and $[f_{P}]$ the isotopy class of $f$ in $D^{2}\setminus P$. We give a non trivial lower bound of the topological entropy for maps in $[f_{P}]$, using the spectral radius of some specializations in $GL(n,\mathbb{C})$ of the Burau matrix associated with $[f_{P}]$ and we discuss some examples.
burau.tex
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% Journal infos --------------------------------------------------
%@ARTICLE{Kolev89,
%  author       = {Kolev, Boris},
%  title        = {Entropie topologique et repr{\'e}sentation de {B}urau},
%  journal      = {C. R. Acad. Sci. Paris S{\'e}r. I Math.},
%  fjournal     = {Comptes Rendus de l'Acad{\'e}mie des Sciences. S{\'e}rie I.
%                  Math{\'e}matique},
%  volume       = 309,
%  year         = 1989,
%  number       = 13,
%  pages        = {835--838},
%  issn         = {0764-4442},
%  coden        = {CASMEI},
%  mrclass      = {58F11 (54C70 57M05)},
%  mrnumber     = {91e:58095},
%  mrreviewer   = {L{\'u}cia Helena Vilas B\^{o}as Mendes},
%}
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    pdftitle={Topological entropy and Burau representation},
    pdfauthor={Boris Kolev}
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\begin{document}

\title{Topological entropy and Burau representation}%
\author{Boris Kolev}%
\address{CMI,39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France}
\email{boris.kolev@cmi.univ-mrs.fr}

\subjclass{20F36 (54C70 57M05)} \keywords{Topological entropy,
Burau representation, Fundamental group, Free differential
calculus}

% ----------------------------------------------------------------
\begin{abstract}
Let $f$ be an orientation-preserving homeomorphism of the disk
$D^{2}$, $P$ a finite invariant subset and $[f_{P}]$ the isotopy
class of $f$ in $D^{2}\setminus P$. We give a non trivial lower
bound of the topological entropy for maps in $[f_{P}]$, using the
spectral radius of some specializations in $GL(n,\mathbb{C})$ of
the Burau matrix associated with $[f_{P}]$ and we discuss some
examples.
\end{abstract}
\maketitle
% ----------------------------------------------------------------
\section{Introduction}

Let $f$ be an orientation-preserving homeomorphism of the disk
$D^{2}$ and $P \subset \interior{D^{2}}$, a finite invariant
subset of $n$ points. $f$ induces a homeomorphism $f_{P}$ on
$D_{P} = D^{2}\setminus P$ and $f_{P}$ induces an automorphism
$f_{P\sharp}$ on $\pi_{1}(D_{P}) \simeq F_{n}$, the free group
with $n$ generators. If $h(f)$ is the \emph{topological entropy}
\cite{AKM65} of $f$ and $\GR{f_{P\sharp}}$ is the \emph{growth
rate} of $f_{P\sharp}$ on $\pi_{1}(D_{P})$, Bowen \cite{Bow78} has
shown that
\begin{equation}\label{BowenInequality}
    h(f) \geq \ln (\GR{f_{P\sharp}}).
\end{equation}

The problem is that in general, it is very hard to compute
$\GR{f_{P\sharp}}$.

By the way, let $f_{P \ast 1}$ be the isomorphism induced by
$f_{P}$ on the first homology group $H_{1}(D_{P};\mathbb{C})$,
then
\begin{equation}\label{ManningInequality}
    \GR{f_{P\sharp}} \geq R(f_{P \ast 1})
\end{equation}
where $R(f_{P \ast 1})$ is the spectral radius of the isomorphism
$f_{P \ast 1}$. But $f_{P \ast 1}$ is just a permutation and so
inequality \eqref{ManningInequality} is always trivial. Therefore,
we cannot by this way detect a positive topological entropy.

It is a well-known fact \cite{Bir74} that the subgroup of
automorphism of $F_{n}$ induced by homeomorphisms of $D_{P}$ is
isomorphic to \emph{Artin's Braid group} $B_{n}$~\footnote{~To be
perfectly exact, this result supposed that we have fixed the star
point of $\pi_{1}(D_{P})$ on the boundary of $D_{P}$ !}. There
exists a famous representation \cite{Bir74} of $B_{n}$ in
$GL(n,\mathbb{Z}[t,t^{-1}])$, the \emph{Burau representation}. In
this paper, we prove the following:

\begin{thm}\label{Main}
Let $\alpha \in B_{n} \subset Aut(F_{n})$ and $B_{\alpha}$ the
Burau matrix of $\alpha$; we have
\begin{equation}\label{KolevInequality}
    \GR{\alpha} \geq \sup \set{R(B_{\alpha}(t));\, t\in \mathbb{C},\, \abs{t} =
    1},
\end{equation}
$R(B_{\alpha}(t))$ being the spectral radius of the matrix
$B_{\alpha}(t) \in GL(n,\mathbb{C})$ when evaluating at a complex
number $t \in \mathbb{C}$.
\end{thm}

\begin{rem}
This result is not really new since it appears not explicitly as a
corollary of results from David Fried in \cite{Fri86}.
Nevertheless, we give here a direct proof of it without any
reference to "twisted" cohomology, using only the definition of
the Burau representation and Fox free differential calculus
\cite{CF77}.
\end{rem}

% ----------------------------------------------------------------

\section{The Braid group and the Burau representation}

\subsection{Artin braid group}

There are several ways to introduce the Braid group $B_{n}$. All
this paragraph is just a short summary of what can be found in
\cite{Bir74}. We define $B_{n}$ in a purely algebraic way as the
subgroup of \emph{right automorphisms} $\alpha$ of the free group
\begin{equation}
F_{n}=\langle x_{1}, x_{2}, \dotsc ,x_{n}\rangle
\end{equation}
which verify:
\begin{align}
    (x_{i})\alpha & = A_{i}x_{\mu_{i}}A_{i}^{-1}, \quad 1 \leq i
    \leq n,\\
    (x_{1}x_{2}\dotsb x_{n})\alpha & = x_{1}x_{2}\dotsb x_{n}
\end{align}
where $(\mu_{1},\mu_{2},\dotsc,\mu_{n})$ is a permutation of
$(1,2, \dotsc , n)$ and $A_{i}$ is an element of $F_{n}$.

This group if of finite type with generators
\begin{equation}\label{generators}
    \sigma_{1}, \sigma_{2}, \dotsc , \sigma_{n-1}
\end{equation}
where:
\begin{equation}
\left\{%
\begin{array}{ll}
    (x_{i})\sigma_{i} & = x_{i}x_{i+1}x_{i}^{-1}, \\
    (x_{i+1})\sigma_{i} & = x_{i},\\
    (x_{j})\sigma_{i} & = x_{j}, \quad j \neq i,i+1.\\
\end{array}%
\right.
\end{equation}
and relations
\begin{align}\label{relations}
\sigma_{i}\sigma_{i+1}\sigma_{i} & =
\sigma_{i+1}\sigma_{i}\sigma_{i+1},\quad
     1 \leq i \leq n-2, \\
\sigma_{i}\sigma_{j} & = \sigma_{j}\sigma_{i},\quad \abs{i-j} \geq
2
\end{align}

\subsection{Free differential calculus}

Let $\mathbb{Z}F_{n}$ be the group ring of $F_{n}$ with integer
coefficients. For $j = 1,2,\dotsc , n$, there exists a
well-defined map:
\begin{equation}\label{FreeDerivative}
    \Dp{x_{j}}: \mathbb{Z}F_{n} \rightarrow \mathbb{Z}F_{n}
\end{equation}
called \emph{free derivative} and verifying:
\begin{align}
    \Dp{x_{j}}(c_{1} + c_{2}) & = \Dp{x_{j}}(c_{1}) +
    \Dp{x_{j}}(c_{2}), \quad c_{1}, c_{2} \in \mathbb{Z}F_{n} ;
    \\
    \Dp{x_{j}}(g_{1} g_{2}) & = \Dp{x_{j}}(g_{1}) +
    g_{1}\Dp{x_{j}}(g_{2}), \quad g_{1}, g_{2} \in F_{n} ;
    \\
    \Dp{x_{j}}(x_{i}) & = \delta_{i,j} \quad 1 \leq i \leq n .
\end{align}

Finally, let $\langle t \rangle$ be the free group with one
generator generated by $t$ and
\begin{equation}\label{projection}
\varphi \left\{%
\begin{array}{lll}
    F_{n} & \rightarrow & \langle t \rangle \\
    x_{i} & \mapsto & t, \quad 1 \leq i \leq n \\
\end{array}%
\right.
\end{equation}
$\varphi$ extends into a morphism $\mathbb{Z}F_{n} \rightarrow
\mathbb{Z}\langle t \rangle = \mathbb{Z}[t, t^{-1}]$ that we will
continue to call $\varphi$.

\subsection{Burau representation}

Consider an element $\alpha$ in $B_{n}$ and define the $n$-square
matrix $B_{\alpha} = (b_{ij})$ by:
\begin{equation}\label{Burau}
    b_{ij} = \varphi \left( \Dp{x_{j}} \big((x_{i})\alpha\big)
    \right) \in \mathbb{Z}[t, t^{-1}] \quad (1 \leq i,j \leq n)
\end{equation}
It is easy to show \cite{CF77} the following relation:
\begin{equation}\label{MatrixProduct}
    B_{\alpha\beta} = B_{\alpha}B_{\beta}.
\end{equation}
The maps $\alpha \mapsto B_{\alpha}$ from $B_{n}$ to $GL(n,
\mathbb{Z}[t,t^{-1}])$ is the \emph{Burau representation} of the
Braid group $B_{n}$.

Let $\M = (\mathbb{Z}[t,t^{-1}])^{n}$ and $V_{1},V_{2}, \dotsc ,
V_{n}$ denote the canonical basis of the free module $\M$. A Burau
matrix $B$ acts (on the right) on raw vectors of $\M$.

Any Burau matrix $B=(b_{ij})$ satisfies
\begin{equation}\label{Burau1}
    \sum_{k}t^{k-1}b_{kj}  = t^{j-1}, \quad (\forall j).
\end{equation}
Hence, the submodule $\mathbf{H}$ of $\M$ defined by
\begin{equation}\label{Subspace}
    X_{1} + X_{2} + \dotsb + X_{n} = 0
\end{equation}
is invariant under $B$. The Burau representation is reducible into
an $(n-1)$-dimensional representation $B^{r}$, called the
\emph{reduced Burau representation}.

\begin{rem}
There is an interesting relation between the Burau representation
of a braid $\alpha$ and the Alexander polynomial, a famous
invariant of links. If we close the braid $\alpha$ to obtain a
link $\widetilde{\alpha}$ and we let
\begin{equation*}
    L = \widetilde{\alpha} \cup m
\end{equation*}
where $m$ is the braid axis (that is a circle which surrounded all
the stings of the braid), $L$ characterizes the link
$\widetilde{\alpha}$ in the 3-sphere $S^{3}$ \cite{Bir74} and the
Alexander polynomial of $L$ is given by (cf.~\cite{Mor85}):
\begin{equation}\label{Morton}
    \Delta (x,t) = \det (B_{\alpha}^{r}(t)-xI_{n-1})
\end{equation}
\end{rem}


\subsection{Spectral properties of Burau matrices}

A Burau matrix $B=(b_{ij})$ satisfies the following relations:
\begin{equation}\label{Burau2}
    \sum_{k}b_{ik}  =1, \quad (\forall i).
\end{equation}
Hence, the raw vector $(1,t,\dotsc , t^{n-1})$ is an eigenvector
for $B$ with eigenvalue $1$. If $P_{B}$ is the characteristic
polynomial of $B$, we have thus:
\begin{equation}\label{Pol1}
P_{B}(X) = \det (XI_{n}-B) = (X-1)P_{B^{r}}(X).
\end{equation}

Squier \cite{Squ84} has shown that the reduced Burau
representation is \emph{unitary}, in the sense that there exits a
non-singular matrix $J \in GL(n-1,\mathbb{Z}[t,t^{-1}])$ such
that:
\begin{equation}\label{Squier}
    B^{r\ast}JB^{r} = J
\end{equation}
where $B^{r\ast}$ is the transpose of the matrix obtained from
$B^{r}$ by exchanging $t$ and $t^{-1}$.

From equation~\eqref{Squier} we deduce that
\begin{equation}\label{Pol2}
    P_{B_{\alpha}^{r}}(1/X) =
    (-1/X)^{n-1}(-t)^{e}P_{B_{\alpha}^{r\ast}}(X),
\end{equation}
where $e$ is the algebraic sum of exponents of $\sigma_{i}$ in
expression of $\alpha$.

Therefore, if we let $t\in \mathbb{C}$, $\abs{t}=1$ then this
relation becomes:
\begin{equation}\label{Pol3}
    P_{B_{\alpha}^{r}}(1/X) =
    (-1/X)^{n-1}(-t)^{e}\overline{P_{B_{\alpha}^{r}}(X)}.
\end{equation}
Hence, if $\lambda$ is an eigenvalue of $B_{\alpha}^{r}$ then
$1/\overline{\lambda}$ is also an eigenvalue of $B^{r}$ with the
same multiplicity.

% ----------------------------------------------------------------

\section{Growth rate of a finite type group automorphism}

Let $G$ be a finite type group, $S = \set{x_{1},x_{2}, \dotsc ,
x_{n}}$ a family of generators for $G$ and $L_{S}(g)$ be the
minimal length of $g\in G$ relatively to $S$, that is the minimum
number of letters $x_{i}$ or $x_{i}^{-1}$ to needed to express
$g$. The growth rate of an automorphism $\alpha : G \rightarrow G$
is defined to be
\begin{equation}\label{GrowthRate}
    \GR{\alpha} = \sup_{g\in G}\set{\limsup_{p \to +\infty} \left( L_{S}
    ((g)\alpha^{p})\right)^{1/p}}.
\end{equation}

It is not hard to see that $\GR{\alpha}$ is independent of the
generating system $S$, that $1 \leq \GR{\alpha} < + \infty$ and
that
\begin{equation}\label{GrowthRateNormal}
\GR{\alpha} = \limsup_{p \to +\infty}\set{\max_{1 \leq i \leq n}
\left( L_{S} ((x_{i})\alpha^{p})\right)^{1/p}}.
\end{equation}

Introducing the \emph{occurrence matrix} $A_{\alpha} = (a_{ij})$
where $a_{ij}$ is the number of occurrence of $x^{\pm 1}$ in the
\emph{reduced word} $(x_{i})\alpha$ and norm $\norm{A_{\alpha}} =
\max_{i}(\sum_{j}a_{ij})$, equation \eqref{GrowthRateNormal}
gives:
\begin{equation}\label{SpectralRadius}
    \GR{\alpha} = \lim_{p \to
    +\infty}\norm{A_{\alpha^{p}}}^{1/p}.
\end{equation}

\begin{rem}
If $G = F_{n}$ is the free group, $S = \set{x_{1},x_{2}, \dotsc ,
x_{n}}$ a system of free generators, $\alpha \in Aut(F_{n})$ and
if moreover there is are no cancellations when we iterate
$\alpha$, then $L_{S}((x_{i}x_{j})\alpha) = L_{S}((x_{i})\alpha) +
L_{S}((x_{j})\alpha)$, for $1 \leq i,j \leq n$. Hence,
$A_{\alpha^{p}} = A_{\alpha}^{p}$, and $\GR{\alpha}$ is just the
spectral radius of matrix $A_{\alpha}$. In the general case, we do
not know any recurrence formula for $A_{\alpha^{p}}$.
\end{rem}

% ----------------------------------------------------------------

\section{Proof of main theorem}

Let's come back to the case where $G = F_{n}$ and $\alpha \in
B_{n}$. Theorem~\ref{Main} is a consequence of preceding
considerations as we shall see now.

Let $\omega =
x_{\mu_{1}^{\varepsilon_{1}}}x_{\mu_{2}^{\varepsilon_{2}}} \dotsm
x_{\mu_{r}^{\varepsilon_{r}}} \in F_{n}$ be a reduced word with
$\varepsilon_{k}= \pm 1$. We have
\begin{equation}\label{DerivativeOfOmega}
    \Dp{x_{j}}(\omega) = \sum_{k=1}^{r}\varepsilon_{k}\delta_{\mu_{k},j}x_{\mu_{1}}^{\varepsilon_{1}} x_{\mu_{2}}^{\varepsilon_{2}}
    \dotsb x_{\mu_{k}}^{(\varepsilon_{k}-1)/2}.
\end{equation}

Therefore, the number of occurrence of $x_{j}^{\pm 1}$ in $\omega$
is equal to the number of monomials in $(\Dp{x_{j}})(\omega)$.
Moreover, if we let
\begin{equation*}
    \varphi \left( (\Dp{x_{j}}(\omega)\right) = \sum a_{n}t^{n},
\end{equation*}
then the number of monomials in $(\Dp{x_{j}})(\omega)$ is greater
than $\sum \abs{a_{n}}$. Now, taking $t \in \mathbb{C}$, $\abs{t}
= 1$, we have
\begin{equation*}
    \abs{ \sum a_{n}t^{n} } \leq \abs{ \sum a_{n} }.
\end{equation*}

Hence, if $\alpha \in B_{n}$, $A_{\alpha}$ is the occurrence
matrix and $B_{\alpha}$ is the Burau matrix, we get for all $t\in
\mathbb{C}$, $\abs{t} = 1$:
\begin{equation*}
    a_{ij} \geq \abs{b_{ij}(t)}.
\end{equation*}

This shows that
\begin{equation*}
    \norm{A_{\alpha^{p}}} \geq \norm{B_{\alpha^{p}}(t)} =
    \norm{B_{\alpha}^{p}(t)},
\end{equation*}
which concluded the proof.


\begin{rem}
For $t=1$, $B_{\alpha}(1)$ is precisely the matrix of the map
induced by $\alpha$ on the homology group
$H^{1}(D_{P};\mathbb{C})$, which is just a permutation matrix.
Hence $t=1$ will always give a trivial result.
\end{rem}

% ----------------------------------------------------------------

\section{Examples}

\subsection{Example 1}
$\alpha = \sigma_{1}\sigma_{2}^{-1} \in B_{3}$.

The action on $F_{3}$ is given by
\begin{equation}\label{Example1}
    \left\{%
\begin{array}{ll}
    (x_{1})\alpha & = x_{1}x_{3}x^{-1}_{1} \\
    (x_{2})\alpha & = x_{1} \\
    (x_{3})\alpha & = x^{-1}_{3}x_{2}x_{3}\\
\end{array}%
\right.
\end{equation}

In this example, there are no cancellations when we iterate
$\alpha$ on each generator $x_{i}$. Hence, if $A_{\alpha^{n}}$ is
the occurrence matrix of $\alpha^{n}$, we have
\begin{equation*}
    A_{\alpha^{n}} = A_{\alpha}^{n}.
\end{equation*}
Therefore, we can compute explicitly $\GR{\alpha}$ as the spectral
radius of $A$ and we find $(3 + \sqrt{5})/2$. By the way, the
characteristic polynomial of $B_{\alpha}$ is given by
\begin{equation*}
    P_{B_{\alpha}}(X) = (X-1)(X^{2}-(1-t-t^{-1})X + 1)
\end{equation*}
For $t=-1$, the root of biggest modulus of $P_{B_{\alpha}}$ is
precisely $(3 + \sqrt{5})/2$. In this case
inequality~\eqref{KolevInequality} is an equality.

This is however not surprising; $\GR{\alpha})$ is always equal to
the spectral radius of $B_{\alpha}(-1)$ if $\alpha \in B_{3}$. Let
$\overline{D}_{P}$ be the \emph{blow-up} of the $3$-punctured disc
(where each puncture is replaced by a circle~\cite{Bow78}). The
isotopy class represented by $\alpha$ in $\overline{D}_{P}$ is
either periodic or pseudo-Anosov (in the Thurston
sense~\cite{Thu88}). In the first case, $\GR{\alpha} =
R(B_{\alpha}) = 1$. In the second case, the foliation of a
pseudo-Anosov map $\phi$ in this class has only singularities on
the boundary of $\overline{D}_{P}$. The two-fold cover of
$\overline{D}_{P}$ that we shall denote $\overline{D}^{2}_{P}$,
known as the hyperbolic involution, is a torus with $4$ holes
(corresponding to each boundary component of $\overline{D}_{P}$).
The lift $\widetilde{\phi}$ of $\phi$ induces an Anosov
diffeomorphism on the torus obtained by replacing each boundary
curve by a point. The matrix $B_{\alpha}^{r}(-1)\in
SL(2,\mathbb{Z})$ is exactly the matrix of a linear Anosov
diffeomorphism $A$ in this class. Hence, the entropy of $\phi$
which is equal to the one of $A$ is equal to $R(B_{\alpha}(-1)$.

\subsection{Example 2}
$\alpha = \sigma_{1}\sigma_{2}^{-1}\sigma_{3}^{-1} \in B_{4}$.

The action on $F_{4}$ is given by
\begin{equation}\label{Example2}
    \left\{%
\begin{array}{ll}
    (x_{1})\alpha & = x_{1}x_{4}x^{-1}_{1} \\
    (x_{2})\alpha & = x_{1} \\
    (x_{3})\alpha & = x^{-1}_{4}x_{2}x_{4} \\
    (x_{4})\alpha & = x^{-1}_{4}x_{3}x_{4} \\
\end{array}%
\right.
\end{equation}

In this example, there are cancellations when we iterate $\alpha$
and so we are not able to compute straightforwardly $\GR{\alpha}$.
The characteristic polynomial of the Burau matrix is:
\begin{equation*}
    P_{B_{\alpha}}(X) =
    (X-1)(X^{3} - (1-t-t^{-1})X^{2} + (t^{-2}-t^{-1}+1)X +
    t^{-1}).
\end{equation*}

For $t=-1$, we get:
\begin{equation*}
    P_{B_{\alpha}}(X) = (X-1)^{3},
\end{equation*}
which gives nothing. But for $t=j=e^{2\pi i/3}$ we obtain
\begin{equation*}
    P_{B_{\alpha}}(X) = X^{3} - 2X^{2} - 2jX + j
\end{equation*}
which has a root of modulus strictly greater than $1$. We can
therefore conclude that $h(f)>0$ for all maps in the corresponding
isotopy class. In fact, in this case this isotopy class is of
pseudo-Anosov type. The growth rate $\GR{\alpha}$ is the greatest
positive root of the polynomial $X^{4} - 2X^{3} - 2X + 1$ (cf.
\cite[appendix A]{Kol91}) which is $\lambda = 1 + \sqrt{3} +
\sqrt{2\sqrt{3}}$ and we will show that this number is neither
obtained by our estimate~\eqref{KolevInequality}.

\begin{lem}
In Example~2, we have:
\begin{equation*}
    \lambda > \sup_{\abs{t}=1} \set{R(B_{\alpha}(t))}.
\end{equation*}
\end{lem}

\begin{proof}
Since $t \mapsto R(B_{\alpha}(t))$ is a continuous function on the
compact set $\abs{t}=1$, if there is equality in
\eqref{KolevInequality} for this $\alpha$, there must exist a
complex number $t$ ($\abs{t}=1$) for which this bound is obtained.
Hence there must exist $t$ ($\abs{t}=1$) such that the polynomial
\begin{equation*}
    \lambda^{3}X^{3} - (1-t-t^{-1})\lambda^{2}X^{2} + (t^{-2}-t^{-1}+1)\lambda X +
    t^{-1})
\end{equation*}
has a root on the unit circle $\abs{X}=1$.

But a necessary condition for a polynomial $P = a_{n}X^{n} +
\dotsb + a_{0}$ ($a_{n}\neq 0$) to have a root on the unit circle
is that $P$ and $Q = \overline{a}_{0}X^{n} + \dotsb +
\overline{a}_{n}$ have a common root. To check this we compute the
resultant of the two polynomials
\begin{gather*}
    \lambda^{3}X^{3} - (1-t-t^{-1})\lambda^{2}X^{2} + (t^{-2}-t^{-1}+1)\lambda X +
    t^{-1} \\
    tX^{3} + (1-t+t^{2})\lambda X^{2} - (1-t-t^{-1})\lambda^{2}X +
    \lambda^{3}
\end{gather*}
and we verify that there is no value of $t$ ($\abs{t}=1$) for
which the resultant vanishes.
\end{proof}

\subsection{Example 3}
$\alpha =
\sigma_{4}\sigma_{3}\sigma_{2}\sigma_{1}\sigma_{4}\sigma_{3} \in
B_{5}$.

The action on $F_{5}$ is given by
\begin{equation}\label{Example3}
    \left\{%
\begin{array}{ll}
    (x_{1})\alpha & = x_{1}x_{2}x^{-1}_{1} \\
    (x_{2})\alpha & = x_{1}x_{3}x_{4}x^{-1}_{3}x^{-1}_{1} \\
    (x_{3})\alpha & = x_{1}x_{3}x_{5}x^{-1}_{3}x^{-1}_{1} \\
    (x_{4})\alpha & = x_{1}x_{3}x^{-1}_{1} \\
    (x_{5})\alpha & = x_{1} \\
\end{array}%
\right.
\end{equation}

This is another interesting example. Here again $\alpha$ represent
a pseudo-Anosov class. The growth rate $\GR{\alpha}$ is the
greatest positive root of $X^{4} - X^{3} - X^{2} - X + 1$ that we
shall call $\lambda$ (cf. \cite[appendix A]{Kol91}).

By the way, we have
\begin{equation*}
    P_{B_{\alpha}(-1)}(X) = (X-1)(X^{4} + X^{3} - X^{2} + X + 1)
\end{equation*}
which has $-\lambda$ as a root. Hence, in this case
\eqref{KolevInequality} is an equality. As in example~1, the
invariant foliation of a pseudo-Anosov representative $\phi$ in
this class has only singularities on the boundary of
$\overline{D}_{P}$. Since $card(P) = 5$ is odd, it can be shown
that the lift $\widetilde{\phi}$ of $\phi$ on the two-fold cover
$\overline{D}_{P}^{2}$  has only singularities on the boundary and
there are of even order \cite{Kol91}. Hence the invariant
foliation of $\widetilde{\phi}$ is transversally orientable and,
$h(\widetilde{\phi}) = R(\widetilde{\phi}_{\ast 1})$. But
$B_{\alpha}^{r}(-1)$ is the homology matrix of $\widetilde{\phi}$
and so
\begin{equation*}
    \lambda = h(\widetilde{\phi}) = R(\widetilde{\phi}_{\ast 1}) =
    R(B_{\alpha}^{r}(-1)).
\end{equation*}
This case of equality will happen each time $card(P)$ is odd and
$\alpha$ represent a pseudo-Anosov class with singularities only
on the boundary of $\overline{D}_{P}$.

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\begin{thebibliography}{1}

\bibitem{AKM65}
R.~L. Adler, A.~G. Konheim, and M.~H. McAndrew, \emph{Topological entropy},
  Trans. Amer. Math. Soc. \textbf{114} (1965), 309--319. \MR{30 \#5291}

\bibitem{Bir74}
Joan~S. Birman, \emph{Braids, links, and mapping class groups}, Princeton
  University Press, Princeton, N.J., 1974, Annals of Mathematics Studies, No.
  82. \MR{51 \#11477}

\bibitem{Bow78}
Rufus Bowen, \emph{Entropy and the fundamental group}, The structure of
  attractors in dynamical systems (Proc. Conf., North Dakota State Univ.,
  Fargo, N.D., 1977), Lecture Notes in Math., vol. 668, Springer, Berlin, 1978,
  pp.~21--29. \MR{80d:58049}

\bibitem{CF77}
Richard~H. Crowell and Ralph~H. Fox, \emph{Introduction to knot theory},
  Springer-Verlag, New York, 1977, Reprint of the 1963 original, Graduate Texts
  in Mathematics, No. 57. \MR{56 \#3829}

\bibitem{Fri86}
David Fried, \emph{Entropy and twisted cohomology}, Topology \textbf{25}
  (1986), no.~4, 455--470. \MR{88k:58074}

\bibitem{Kol91}
Boris Kolev, \emph{Dynamique topologique en dimension 2: Orbites
  p{\'e}riodiques et entropie topologique}, Ph.D. thesis, Universit{\'e} de
  Nice, France, 1991.

\bibitem{Mor85}
H.~R. Morton, \emph{Exchangeable braids}, Low-dimensional topology (Chelwood
  Gate, 1982), London Math. Soc. Lecture Note Ser., vol.~95, Cambridge Univ.
  Press, Cambridge, 1985, pp.~86--105. \MR{87e:57010}

\bibitem{Squ84}
Craig~C. Squier, \emph{The {B}urau representation is unitary}, Proc. Amer.
  Math. Soc. \textbf{90} (1984), no.~2, 199--202. \MR{85b:20056}

\bibitem{Thu88}
William~P. Thurston, \emph{On the geometry and dynamics of diffeomorphisms of
  surfaces}, Bull. Amer. Math. Soc. (N.S.) \textbf{19} (1988), no.~2, 417--431.
  \MR{89k:57023}

\end{thebibliography}
