# Volume growth for infinite graphs and translation surfaces

P. Colognese and M. Pollicott Department of Mathematics, Warwick University,
Coventry, CV4 7AL, UK

( Warwick University
)

##  1 Introduction

We begin by recalling the definition of volume entropy for compact Riemannian
manifolds due to Manning [7]. Let $M$ be a compact manifold with Riemannian
metric $\rho$ and universal cover $\widetilde{M}$ equipped with the lifted
metric $\widetilde{\rho}$. Fix a point $c\in\widetilde{M}$ and consider a ball
$B(c,R)$ of radius $R>0$ centred at $c$.

######  Definition 1.1.

The volume entropy of $M$ is defined by

$h=h(M):=\lim_{R\to\infty}\frac{1}{R}\log\hbox{\rm
Vol}_{\tilde{\rho}}(B(c,R)),$

where $\hbox{\rm Vol}_{\tilde{\rho}}$ denotes the Riemannian volume on
$\widetilde{M}$ with respect to $\widetilde{\rho}$.

For manifolds $(M,\rho)$ of non-positive curvature this coincides with the
topological entropy $h$ of the associated geodesic flow [7]. In the case of
manifolds with negative sectional curvature, Margulis [9] showed in his thesis
that there is a simple asymptotic formula: There exists $C>0$ such that

$\lim_{R\to+\infty}\frac{\hbox{\rm Vol}_{\tilde{\rho}}(B(c,R))}{e^{hR}}=C.$

A closely related result in [9] gave an asymptotic formula for the number
$\Pi(x,R)$ of geodesic arcs starting and finishing at a given point $x$ of
length at most $R$: There exists $D>0$ such that

$\lim_{R\to+\infty}\frac{\Pi(x,R)}{e^{hR}}=D.$

A related notion of volume entropy was considered for directed, finite,
connected, non-cyclic graphs without terminal vertices by Lim in [6]. In this
note we extend Lim’s definition of volume entropy to suitable infinite graphs
and show the analogue of Margulis’ result in this context (Theorem 2.4). As an
application we show a version of Margulis’ theorem for the natural analogue of
volume growth for translation surfaces (Theorem 6.11).

This note originated as a summer MPhil project of the first author. It may
have been possible to apply the transfer operator methods in [15], but instead
we employ a more direct and elementary approach.

We are grateful to A. Eskin, J. Chaika, R. Sharp, S. Ghazouani and the three
anonymous referees for their useful comments.

##  2 Infinite Graphs

In this section we will introduce the types of graphs we shall we working with
as well as basic definitions which will be used throughout the paper.

Let $\mathcal{G}$ be a non-empty connected oriented graph. Let
$\mathcal{V}=\mathcal{V}(\mathcal{G})$ and
$\mathcal{E}=\mathcal{E}(\mathcal{G})$ be the vertex and oriented edge sets
respectively. For every edge $e$, let $i(e)$ and $t(e)$ denote the initial and
the terminal vertex of $e$, respectively. We can define a length distance $d$
on $\mathcal{G}$ by introducing a length function
$\ell:\mathcal{E}\to\mathbb{R}$ which assigns a positive real number $\ell(e)$
to each edge $e\in\mathcal{E}$.

######  Example 2.1 (Infinite Graph).

Consider a graph $\mathcal{G}$ formed from one vertex and a countably infinite
number of edges.

$\cdots$$v$$e_{1}$$e_{2}$$e_{3}$$e_{4}$ Figure 1:  A single vertex
$\mathcal{V}=\\{v\\}$ and infinitely many edges
$\mathcal{E}=\\{e_{n}\\}_{n=1}^{\infty}$.

A path in $\mathcal{G}$ corresponds to a sequence of edges $p=e_{1}\ldots
e_{n}$ for which $t(e_{j})=i(e_{j+1})$, for $1\leq j<n$ and we denote its
length by $\ell(p)=\sum_{j=1}^{n}\ell(e_{j})$.

Let $\mathcal{P}_{{\mathcal{G}}}(x,R)=\\{p=e_{1}\ldots e_{n}\hbox{ :
}i(e_{1})=x,\ell(p)\leq R\\}$ denote the set of all such paths of length at
most $R$ starting at $x\in\mathcal{V}({\mathcal{G}})$. We denote its
cardinality by $N_{{\mathcal{G}}}(x,R)=\\#\mathcal{P}_{{\mathcal{G}}}(x,R)$.

######  Definition 2.2.

We define the volume entropy of $(\mathcal{G},\ell,x)$ as

$h(\mathcal{G},\ell,x)=\limsup_{R\rightarrow+\infty}\frac{1}{R}\log
N_{\mathcal{G}}(x,R).$

However, we need to make further assumptions on the length function $\ell$ for
$h(\mathcal{G},\ell,x)$ to be finite. To see this, consider the graph
$\mathcal{G}$ in Example 2.1 which has a single vertex and an infinite number
of edges, and assume that the lengths don’t tend to infinity. Then for $R$
sufficiently large, $N_{\mathcal{G}}(x,R)=\infty$ and thus
$h(\mathcal{G},\ell,x)=\infty$.

We summarise below the properties of the graph that are needed in the proof.

Graph Hypotheses. Henceforth, we shall consider graphs with finite vertex set
$\mathcal{V}$ and a countable edge set $\mathcal{E}$. Furthermore we require
that $\mathcal{E}$ and the associated length function satisfy the following
properties:

  1. (H1)

For all $\sigma>0$ we have $\sum_{e\in\mathcal{E}}e^{-\sigma\ell(e)}<\infty$;

  2. (H2)

For all edges $e,e^{\prime}\in\mathcal{E}$ there exists a path in
$\mathcal{G}$ which starts with $e$ and ends with $e^{\prime}$111A slightly
weaker assumption would be to require that for a sufficiently large finite
subset $\mathcal{E}_{0}\subset\mathcal{E}$, for every
$e,e^{\prime}\in\mathcal{E}_{0}$ there exists a path in $\mathcal{G}$ which
starts with $e$ and ends with $e^{\prime}$; and

  3. (H3)

There does not exist a $d>0$ such that

$\\{\ell(c)\hbox{ : }c\hbox{ is a closed path}\\}\subset d\mathbb{N}.$

Under the above hypotheses, the volume entropy $h=h(\mathcal{G},\ell,x)$ does
not depend on the choice of base point $x$.

######  Lemma 2.3.

If the graph $\mathcal{G}$ satisfies (H1) and (H2) then $0<h<\infty$.

###### Proof.

By assumption (H2), and the pigeonhole principle applied to $\mathcal{V}$,
there exist a path connecting the base point $x$ to some vertex $v$ and two
closed paths, $c_{1}$ and $c_{2}$, which pass through $v$. By considering all
possible concatenations of these closed paths it is clear that there exists
$b>0$ such that $N_{\mathcal{G}}(x,R)\geq 2^{\lfloor R/b\rfloor}$ for all
$R>0$ and hence $h\geq\frac{\log 2}{b}>0$.

To see that $h$ is finite we can formally write

$\sum_{p\in\mathcal{P}_{\mathcal{G}}(x,R)}e^{-\sigma\ell(p)}\leq\sum_{n=1}^{\infty}\bigg{(}\sum_{e\in\mathcal{E}}e^{-\sigma\ell(e)}\bigg{)}^{n},$
$None$

for $\sigma>0$, where the Right Hand Side involves all possible sums of edge
lengths. Using (H1) one can see that for $\sigma=\sigma_{0}$ sufficiently
large $\sum_{e\in\mathcal{E}}e^{-\sigma\ell(e)}<1$ and thus the geometric
series on the Right Hand Side of (2.1) converges. In particular, since $h$ is
easily seen to be the absicssa of convergence of the series on the Left Hand
Side of (2.1) we see that $h\leq\sigma_{0}<+\infty$, as required. ∎

Our main result for $\mathcal{G}$ is the following asymptotic for the growth
of paths.

######  Theorem 2.4.

If the graph $\mathcal{G}$ satisfies (H1),(H2) and (H3) then there exists a
constant $C>0$ such that $N_{{\mathcal{G}}}(x,R)\sim Ce^{hR}$, i.e.,

$\lim_{R\to+\infty}\frac{N_{{\mathcal{G}}}(x,R)}{e^{hR}}=C.$

The proof follows the lines of the classical proof of the prime number
theorem. In particular, it is based on the use of a Tauberian theorem (in
Section 5). This, in turn, depends on the properties of the complex function
$\eta_{\mathcal{G}}(z)$, the Laplace transform of $N_{{\mathcal{G}}}(x,R)$
(defined in Section 4). The function $\eta_{\mathcal{G}}(z)$ is analysed using
matrices introduced in the next section. In the special case of finite graphs,
the asymptotic in Theorem 2.4 could be easily deduced using ideas in [13] for
finite matrices.

######  Remark 2.5.

Without hypothesis (H3) this theorem may not hold. For example, even in the
case of finite graphs, if we consider the graph $\mathcal{G}$ with a single
vertex and two edges of length 1, then $N_{\mathcal{G}}(x,R)=2^{\lfloor
R\rfloor}$ for all $R>0$. In this case, the limit in Theorem 2.4 does not
converge.

##  3 Countable Matrices

In this section consider a graph $\mathcal{G}$ and length function $\ell$
which satisfy hypotheses (H1)-(H3). Let us order the edge set
$\mathcal{E}=(e_{a})_{a\in\mathbb{N}}$ by non-decreasing length and write
$\ell(a):=\ell(e_{a})$, $a\in\mathbb{N}$.

######  Definition 3.1.

We can associate to $\mathcal{G}$ the infinite matrix $M_{0}$ defined by

$M_{0}(a,b)=\begin{cases}1&\mbox{if }t(a)=i(b),\\\ 0&\mbox{otherwise.
}\end{cases}$

For each $z\in\mathbb{C}$ we define the matrix $M_{z}$ by
$M_{z}(a,b)=M_{0}(a,b)e^{-z\ell(b)}$ for $a,b\in\mathcal{E}$.

Let $P(n,a,b)$ denote the set of paths in $\mathcal{G}$ consisting of $n$
edges, starting with edge $e_{a}$ and ending with edge $e_{b}$. It then
follows from formal matrix multiplication that for any $n\geq 1$, we can write
the $(a,b)^{th}$ entry of the $n^{th}$ power of the matrix as:

$M_{z}^{n}(a,b)=e^{z\ell(a)}\sum_{p\in P(n+1,a,b)}e^{-z\ell(p)},$ $None$

which will be finite by hypothesis (H1).

Given a matrix $L=\left(L(a,b)\right)_{a,b=1}^{\infty}$ with
$\sup_{a}\sum_{b}|L(a,b)|<+\infty$ we can associate to $L$ a bounded linear
operator $\widehat{L}:\ell^{\infty}(\mathbb{C})\to\ell^{\infty}(\mathbb{C})$
by

$\widehat{L}(\underline{u})=\Big{(}\sum_{b=1}^{\infty}L(a,b)u_{b}\Big{)}_{a=1}^{\infty}\hbox{
where }\underline{u}=(u_{b})_{b=1}^{\infty}\in\ell^{\infty}(\mathbb{C}).$

In particular, by hypothesis (H1), when $Re(z)>0$ we can associate to $M_{z}$
a bounded operator
$\widehat{M_{z}}:\ell^{\infty}(\mathbb{C})\to\ell^{\infty}(\mathbb{C})$ by

$\widehat{M_{z}}(\underline{u})=\Big{(}\sum_{b=1}^{\infty}M_{z}(a,b)u_{b}\Big{)}_{a=1}^{\infty}.$

To proceed, we would like to understand the domain of meromorphicity of the
linear operator
$(I-\widehat{M_{z}})^{-1}:\ell^{\infty}(\mathbb{C})\to\ell^{\infty}(\mathbb{C})$,
where $I$ denotes the identity operator. To this end, we shall make use of an
idea by Hofbauer and Keller in [4], where they observe that the invertibility
of certain operators of the above form depends only on the determinant of an
associated finite matrix.

Fix $\epsilon>0$ and, for convenience, assume also $h>\epsilon$. Given $k\geq
1$, we can truncate the matrix $M_{z}$ to the $k\times k$ matrix
$A_{z}=(M_{z}(i,j))_{i,j=1}^{k}$. Then we can then write

$M_{z}=\left(\begin{matrix}A_{z}&B_{z}\\\ C_{z}&D_{z}\end{matrix}\right)$

where, in particular, $D_{z}=\left(M_{z}(i+k,j+k)\right)_{i,j=1}^{\infty}$.
Again, we can interpret $I-\widehat{D_{z}}$ as a bounded linear operator on
$\ell^{\infty}(\mathbb{C})$ and write
$(I-\widehat{D_{z}})^{-1}=\sum_{m=0}^{\infty}\widehat{D_{z}}^{m}$ if the
operator $\widehat{D_{z}}$ has norm $\|\widehat{D_{z}}\|<1$. In particular,
this is true when $Re(z)\geq\epsilon$ for $k$ sufficiently large, since by
(H1) we have

$\|\widehat{D_{z}}\|\leq\sup_{n\in\mathbb{N}}\sum_{m=1}^{\infty}|D_{z}(n,m)|\leq\sum_{m=1}^{\infty}e^{-Re(z)\ell(m+k)}\leq\sum_{m=1}^{\infty}e^{-\epsilon\ell(m+k)}<1.$
$None$

Writing $\ell^{\infty}(\mathbb{C})$ as the corresponding direct sum of two
subspaces, we can then easily verify that

$I-\widehat{M_{z}}=\left(\begin{matrix}I-\widehat{A_{z}}-\widehat{B_{z}}(I-\widehat{D_{z}})^{-1}\widehat{C_{z}}&-\widehat{B_{z}}(I-\widehat{D_{z}})^{-1}\\\
0&I\end{matrix}\right)\left(\begin{matrix}I&0\\\
-\widehat{C_{z}}&I-\widehat{D_{z}}\end{matrix}\right).$ $None$

Let us denote the $k\times k$ matrix $W_{z}:=A_{z}+B_{z}(I-D_{z})^{-1}C_{z}$,
where each entry is given by a convergent series. By (3.3), whenever
$\det(I-W_{z})\neq 0$ then we see that $I-\widehat{M_{z}}$ is invertible, with
inverse

$(I-\widehat{M_{z}})^{-1}=\left(\begin{matrix}I&0\\\
(I-\widehat{D_{z}})^{-1}\widehat{C_{z}}&(I-\widehat{D_{z}})^{-1}\end{matrix}\right)\left(\begin{matrix}(I-\widehat{W_{z}})^{-1}&(I-\widehat{W_{z}})^{-1}\widehat{B_{z}}(I-\widehat{D_{z}})^{-1}\\\
0&I\end{matrix}\right).$ $None$

This leads to the following result.

######  Lemma 3.2.

The operator $(I-\widehat{M_{z}})^{-1}$ has an analytic extension to $Re(z)>0$
except when $\det(I-W_{z})=0$.

###### Proof.

This follows from the identity (3.4) and since the $\epsilon>0$ chosen in the
above construction can be chosen arbitrarily small. ∎

##  4 Complex functions

We can now introduce a complex function whose analytic properties will be
useful in deriving our asymptotic estimates for $N_{\mathcal{G}}(x,R)$. Fix
$x\in\mathcal{V}$.

######  Definition 4.1.

We can formally define the complex function

$\eta_{\mathcal{G}}(z)=\int_{0}^{\infty}e^{-zR}dN_{\mathcal{G}}(x,R)=\sum_{p\in
P(x)}e^{-z\ell(p)},\quad z\in\mathbb{C},$

where $P(x)=\\{p=e_{1}\cdots e_{n}\hbox{ : }n\geq 0,i(e_{1})=x\\}$ is the set
of paths in $\mathcal{G}$ starting at $x$.

We first observe that $\eta_{\mathcal{G}}(z)$ converges to an analytic
function for $Re(z)>h$, by virtue of Definition 2.2. In order to construct a
meromorphic extension of $\eta_{\mathcal{G}}(z)$ we shall relate
$\eta_{\mathcal{G}}(z)$ to the matrix $M_{z}$. For $Re(z)>0$, we define:

  1. (a)

$\underline{w}(z)=(\chi_{\mathcal{E}_{x}}(e_{j})e^{-z\ell(j)})_{j=1}^{\infty}\in\ell^{1}(\mathbb{C})$
where $\chi_{\mathcal{E}_{x}}$ denotes the characteristic function of the set
$\mathcal{E}_{x}=\\{e\in\mathcal{E}\hbox{ : }i(e)=x\\}$ of edges whose initial
vertex is $x$; and

  2. (b)

$\underline{1}=(1)_{j=1}^{\infty}\in\ell^{\infty}(\mathbb{C})$ is the vector
all of whose entries are equal to $1$,

then we can formally rewrite $\eta_{\mathcal{G}}(z)$ as

$\displaystyle\eta_{\mathcal{G}}(z)=\sum_{p\in P(x)}e^{-z\ell(p)}$
$\displaystyle=\underline{w}(z)\cdot\Big{(}\sum_{n=0}^{\infty}\widehat{M_{z}}^{n}\Big{)}\underline{1}$
$None$
$\displaystyle=\underline{w}(z)\cdot\Big{(}I-\widehat{M_{z}}\Big{)}^{-1}\underline{1},$

where $w\cdot v=\sum_{j=1}^{\infty}w_{j}v_{j}$ for $w\in\ell^{1}(\mathbb{C})$
and $v\in\ell^{\infty}(\mathbb{C})$. Observe that for $Re(z)>0$ we have
$\underline{w}(z)\in\ell^{1}(\mathbb{C})$ by (H1). In particular, by Lemma 3.2
the expression in (4.1) extends to $Re(z)>0$, and the locations of the poles
are given by those $z$ such that the finite rank operator $(I-W_{z})$ is not
invertible. Moreover, we can easily write

$\eta_{\mathcal{G}}(z)=\frac{\phi(z)}{\det(I-W_{z})}$ $None$

where $\phi(z)$ is holomorphic on $Re(z)>0$.

######  Proposition 4.2.

$\eta_{\mathcal{G}}(z)$ has a meromorphic extension to $Re(z)>0$.

###### Proof.

Observe that $\det(I-W_{z})$ is the sum of a countable number of holomorphic
functions which uniformly converge on any compact domain in $Re(z)>0$ and
hence $\det(I-W_{z})$ is holomorphic. The result follows from the identity
(4.2). ∎

Let $\epsilon<h$. By (3.2) we can choose $k$ large enough such that
$(I-\widehat{D_{z}})$ is invertible, on the half plane $Re(z)\geq\epsilon$.
Recall that a non-negative $n\times n$ matrix $M$ is irreducible if for all
$i,j$ satisfying $1\leq i,j\leq n$ there exists a natural number $m$ such that
$(M^{m})_{i,j}>0$.

######  Lemma 4.3.

Let $\sigma>0$. Then $W_{\sigma}$ is a non-negative irreducible matrix.
Furthermore, $W_{\sigma}$ has a simple maximal positive eigenvalue
$\rho(\sigma)=\rho(W_{\sigma})$, which depends analytically on $\sigma$ and
satisfies $\rho^{\prime}(\sigma)<0.$

###### Proof.

Recall that $W_{\sigma}=A_{\sigma}+B_{\sigma}(I-D_{\sigma})^{-1}C_{\sigma}$,
which by construction is a non-negative matrix. We can also deduce that the
matrix $W_{\sigma}$ is irreducible. To see this, note that by assumption (H2),
for all $1\leq i,j\leq k$, there exists some path of length $n$ starting with
edge $e_{i}$ and ending with edge $e_{j}$. Such a path can be broken up into
sub-paths of two types. The first type consists of those paths that stay
completely within $\\{e_{1},\cdots,e_{k}\\}$, and the second type which
consists of those paths that initially enter the complement
$\mathcal{E}-\\{e_{1},\cdots,e_{k}\\}$ and finally leave at their end. Note
that $W_{\sigma}^{n}(i,j)$ is a sum including powers of $A_{\sigma}(i,j)$
(corresponding to sub-paths of the first type) and
$B_{\sigma}(I-D_{\sigma})^{-1}C_{\sigma}$ (corresponding to sub-paths of the
second type), where the powers are less than or equal to $n$. Hence
$W^{n}_{\sigma}(i,j)>0$.

We can now apply the Perron-Frobenius theorem (see [3]) to deduce that the
maximal positive eigenvalue $\rho(\sigma)>0$ for $W_{\sigma}$ exists and that
$W_{\sigma}$ has associated positive left and right eigenvectors $u(\sigma)$
and $v(\sigma)$ (which we normalise so that $u(\sigma)v(\sigma)=1$). By
differentiating the eigenvalue equations for $u(\sigma)$ and $v(\sigma)$, one
can show that

$\rho^{\prime}(\sigma)=u(\sigma)^{T}W^{\prime}_{\sigma}v(\sigma)<0,$

where $W^{\prime}_{\sigma}$ is the matrix with entries
$W^{\prime}_{\sigma}(i,j)=\frac{dW_{x}(i,j)}{dx}(\sigma)<0$ for all $i,j$ (see
[14] for a similar argument). ∎

######  Proposition 4.4.

$h$ is a simple pole of $\eta_{\mathcal{G}}(z)$.

###### Proof.

For $z$ in a neighbourhood of $h$, we denote by $\rho(z)$ the perturbed
eigenvalue of $W_{z}$. We can write
$\det(I-W_{z})=(1-\rho(z))\Pi_{i=2}^{k}(1-\lambda_{i}(z))$, where the
$\lambda_{i}(z)$ denote the other eigenvalues of $W_{z}$. Since the
$\lambda_{i}(z)$ are bounded away from 1 for $z$ near $h$ (by the Perron-
Frobenius theorem and standard perturbation theory), $\phi(h)\neq 0$ and
$\rho^{\prime}(h)\neq 0$ (by Lemma 4.3), we can conclude that
$(z-h)\eta_{\mathcal{G}}(z)$ converges to a non-zero constant, as $z$ tends to
$h$. ∎

######  Proposition 4.5.

$\eta_{\mathcal{G}}(z)$ has no poles other than $h$ on the line $Re(z)=h$.

###### Proof.

Suppose for a contraction that there exists another pole at $h+it$ ($t\neq
0$). Let $c$ be any closed path and choose an integer $k_{c}>k$ such that the
edges of $c$ have index smaller than $k_{c}$. Then construct the $k_{c}\times
k_{c}$ matrices $W_{z}$. From equation (4.2) we see that $\det(I-W_{h+it})=0$,
and thus $1$ is an eigenvalue for $W_{h+it}$ and $W_{h}$. Furthermore, we can
see that $\rho(W_{h})=1$ since otherwise $\eta_{\mathcal{G}}(z)$ has a pole at
$c>h$, contradicting Definition 2.2.

Next observe that $|W_{h+it}(a,b)|\leq W_{h}(a,b)$ for all $1\leq a,b\leq k$.
Since $\rho(W_{h+it})\geq 1=\rho(W_{h})$, we can apply Wielandt’s theorem (see
[3]) which allows us to conclude that $\rho(W_{h+it})=\rho(W_{h})=1$ and that
there exists a diagonal matrix $D$, whose non-zero entries have unit modulus
such that $W_{h+it}=DW_{h}D^{-1}$, and thus for all $n$ we have
$W_{h+it}^{n}=DW_{h}^{n}D^{-1}$.

Suppose that the closed path $c$ begins with some edge $e_{a}$ and consists of
$n$ edges. One can check that $W_{h+it}^{n}(a,a)=W_{h}^{n}(a,a)$ (since
$W_{h+it}^{n}=DW_{h}^{n}D^{-1}$) and that $e^{(h+it)\ell(c)}$ is one of the
terms in the left hand sum. However, this can only hold if $t$ is such that
$\ell(c)t=2\pi m_{c}$ for some non-zero integer $m_{c}$. As $c$ was arbitrary,
the above construction implies that for all closed paths $c$, $\ell(c)\in
d\mathbb{N}$ with $d=2\pi/t$ which contradicts (H3). ∎

##  5 Proof of Theorem 2.4

We can complete the proof using a similar approach to Parry in [13], where he
considered only finite matrices. In particular, we will use the following
formulation of the Ikehara–Wiener Tauberian theorem [2] applied to our
counting function, $N_{\mathcal{G}}(x,R)$.

######  Theorem 5.1 (Ikehara–Wiener Tauberian theorem).

Let $A:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a monotonic, non-decreasing
function and formally denote $\eta(z):=\int_{0}^{\infty}e^{-zR}dA(R)$, for
$z\in\mathbb{C}$. Then suppose that $\eta(z)$ has the following properties:

  1. 1.

there exists some $a>0$ such that $\eta(z)$ is analytic on $Re(z)>a$;

  2. 2.

$\eta(z)$ has a meromorphic extension to a neighbourhood of the half-plane
$Re(z)\geq a$;

  3. 3.

$a$ is a simple pole for $\eta(z)$, i.e., $C=\lim_{\epsilon\searrow
0}(z-a)\eta(z)>0$; and

  4. 4.

the extension of $\eta(z)$ has no poles on the line $Re(z)=a$ other than $a$.

Then $A(R)\sim Ce^{aR}$ as $R\rightarrow+\infty$.

From the results in the previous section we see that the
$\eta_{\mathcal{G}}(s)$ satisfies the assumptions of Theorem 5.1 with $a=h$
and so we have proved Theorem 2.4.

##  6 Translation surfaces

In this section we will consider a definition of volume entropy for
translation surfaces and prove asymptotic results using the work developed in
the previous sections.

######  Definition 6.1.

A translation surface $X$ is a compact surface with a flat metric except at a
finite set $\Sigma=\\{x_{1},\ldots,x_{n}\\}$ of singular points with cone
angles $2\pi(k(x_{i})+1)$, where $k(x_{i})\in\mathbb{N}$, for $i=1,\ldots,n$.

A path which does not pass through singularities is a locally distance
minimizing geodesic if it is a straight line segment. This includes geodesics
which start and end at singularities, known as saddle connections. We will
consider oriented saddle connections.

Geodesics can change direction if they go through a singular point, and a pair
of line segments ending and beginning, respectively, at a singular point form
a geodesic if the angle between them is at least $\pi$. Thus a locally
distance minimising geodesic (of length $R$) on a translation surface $X$ with
singularity set $\Sigma$, is a curve $\gamma:[0,R]\rightarrow X$ satisfying
the following conditions:

  * •

There exist $0\leq t_{1}<...<t_{n}\leq R$, where $n\geq 0$, such that
$\gamma(t_{i})\in\Sigma$;

  * •

For $t_{i}<t<t_{i+1}$ $\gamma(t)\in X\backslash\Sigma$;

  * •

$\gamma:(t_{i},t_{i+1})\rightarrow X\backslash\Sigma$ is a geodesic segment
(possibly a saddle connection);

  * •

The smallest angle between $\gamma|_{(t_{i-1},t_{i})}$ and
$\gamma|_{(t_{i},t_{i+1})}$ is at least $\pi$ (cf. [1], Lemma 2.1).

Let ${\mathcal{S}}=\\{s_{1},s_{2},...\\}$ be the set of oriented saddle
connections ordered by non-decreasing lengths.

######  Definition 6.2.

We define a saddle connection path $p=(s_{i_{1}},...,s_{i_{n}})$ to be a
finite string of oriented saddle collections $s_{i_{1}},...,s_{i_{n}}$ which
form a geodesic path.

We denote by $\ell(p)=\ell(s_{1})+\ell(s_{2})+\cdots+\ell(s_{n})$ the sum of
the lengths of the constituent saddle connections. We let $i(p),t(p)\in\Sigma$
denote the initial and terminal singularities, respectively, of the saddle
connection path $p$.

######  Example 6.3 (Square tiled surfaces [16]).

We can consider the square-tiled surfaces by identifying opposite sides of
arrangements of a finite number of copies of the unit square (Figure 2). The
values of the lengths of the saddle connections are of the form
$\\{\sqrt{n^{2}+m^{2}}\hbox{ : }(n,m)\in\mathbb{Z}^{2}-(0,0)\hbox{ co-
prime}\\}$.

Figure 2: (i) A square tiled surface formed from four tiles; (ii) a square
tiled surface formed from three tiles.

We now turn our attention to defining a notion of volume entropy for
translation surfaces in terms of the growth of the volume of a ball as its
radius tends to infinity. By analogy with the definition of volume entropy for
Riemannian manifolds (Definition 1.1) we can consider the rate of growth of
balls in the universal cover $\widetilde{X}$ of $X$.

######  Definition 6.4.

Let $\tilde{x}\in\widetilde{X}$ and consider a ball
$B(\tilde{x},R)\subset\widetilde{X}$ of radius $R>0$ with centre
$\widetilde{x}$. We define the volume entropy of $X$ to be

$h=h(X):=\limsup_{R\rightarrow+\infty}\frac{1}{R}\log\hbox{\rm
Vol}_{\widetilde{X}}(B(\tilde{x},R))$

where $\hbox{\rm Vol}_{\widetilde{X}}$ denotes the natural volume on
$\widetilde{X}$.

Definition 6.4 is closely related to the definition of Dankwart [1], which was
formulated in terms of orbital counting. As in the case of the definitions of
volume entropy for Riemannian manifolds and finite metric graphs, $h$ is
independent of the choice of $\tilde{x}$. For convenience, we can take
$\tilde{x}$ to be the lift of a singularity $x\in\Sigma$.

It is useful to interpret this definition in terms of $X$ rather than
$\widetilde{X}$. To this end we have the following definition.

######  Definition 6.5.

Let $m_{R}(y)$ be number of distinct geodesic arcs in $X$ from $x$ to $y$ of
length at most $R$.

We can now rewrite $\hbox{\rm
Vol}_{\widetilde{X}}(B(\tilde{x},R))=\int_{X}m_{R}(y)d\hbox{\rm Vol}_{X}(y)$
(see Figure 3). For economy of notation we will write $V(x,R):=\hbox{\rm
Vol}_{\widetilde{X}}(B(\tilde{x},R))$.

$4$$2$$2$$2$$2$$3$$3$$3$$3$

Figure 3: (i) A small ball centred at a singularity; (ii) As the radius $R$
increases the ball overlaps with itself (and the values of the multiplicity
function $m_{R}(\cdot)$ are indicated).

$R$

Figure 4: (i) The radii of $B(x,R)$ are concatenations of saddle connections
followed by a radial line segment from a singularity; (ii) A heuristic figure
illustrating that the boundary of $B(x,R)$ will consist of the union of
circular arcs centred on singularities reached via concatenations of saddle
connections

Let $x\in\Sigma$ be a singularity, then we define

$\pi(x,R):=\left\\{p\hbox{ : }i(p)=x\hbox{ and }l(p)\leq R\right\\}$

to be the number of saddle connection paths starting at $x$ of length less
than or equal to $R$.

######  Lemma 6.6.

Let $X$ be a translation surface and fix a singularity $x\in\Sigma$ and let
$2\pi(k(x)+1)$ be the cone angle of $x$. Then for $R>0$,

$V(x,R)=(k(x)+1)\pi R^{2}+\sum_{p\in\pi(x,R)}k(t(p))\pi(R-\ell(p))^{2},$

where the singularity at the end of path $p$ has cone angle $2\pi(k(t(p))+1)$.

###### Proof.

The volume contributed by the geodesics starting from $x$ which do not pass
through a singularity is given by $(k(x)+1)\pi R^{2}$, where $2\pi(k(x)+1)$ is
the cone angle at $x$. On the other hand, the contribution to the volume by
those geodesics $\gamma$ which pass through one or more singularities comes
when the geodesic leaves its last singularity at time $\ell({p})<R$, say. It
can exit in one of $2\pi k(p)$ directions. Then the total volume of such
$\gamma$ is given by $k(t(p))\pi(R-(\ell(p))^{2}$. ∎

We shall now prove asymptotic results for translation surfaces using the
analysis developed for infinite graphs.

######  Definition 6.7.

We can associate to $X$ the countable matrix $M_{0}$, indexed by
$\mathcal{S}$, defined by

$M_{0}(s,s^{\prime})=\begin{cases}1&\mbox{if }\text{ss${}^{\prime}$ form a
saddle connection path},\\\ 0&\mbox{otherwise. }\end{cases}$

For each $z\in\mathbb{C}$ we define the matrix $M_{z}$ by
$M_{z}(s,s^{\prime})=M_{0}(s,s^{\prime})e^{-z\ell(s^{\prime})}$ for
$s,s^{\prime}\in\mathcal{S}$.

In order that the matrices have the same properties that served us well for
graphs, we require specific features of a translation surface.

Translation Hypotheses. Henceforth, we shall consider translation surfaces
whose countable set of saddle connections is denoted by $\mathcal{S}$.
Moreover, we require that $\mathcal{S}$ and the lengths of the saddle
connections satisfy the following properties:

  1. (T1)

For all $\sigma>0$ we have $\sum_{s\in\mathcal{S}}e^{-\sigma\ell(s)}<+\infty$;

  2. (T2)

For any directed saddle connections $s,s^{\prime}\in\mathcal{S}$ there exists
a saddle connection path beginning with $s$ and ending with $s^{\prime}$; and

  3. (T3)

There does not exist a $d>0$ such that

$\\{\ell(c)\hbox{ : }c\hbox{ is a closed saddle connection path}\\}\subset
d\mathbb{N}.$

We claim that the above hypotheses hold for all translation surfaces.

Property (T1) follows from the lower bound in following result (see [10] and
[11]).

######  Proposition 6.8.

Let $X$ be a translation surface and let $N(X,L)$ denote the number of saddle
connections on $X$ of length less than or equal to $L$. Then there exists
constants $0<c_{1}<c_{2}<\infty$ such that

$c_{1}L^{2}\leq N(X,L)\leq c_{2}L^{2},$

for $L$ sufficiently large.

To see that Hypotheses (T2) and (T3) hold for all translation surfaces, we
require the following result in [1] which we restate for our purposes here.

######  Proposition 6.9.

Let $X$ be a translation surface. If $s,s^{\prime}\in\mathcal{S}$ are oriented
saddle connections then there exists a saddle connection path which starts
with $s$ and ends with $s^{\prime}$.

Hypothesis (T2) follows immediately from this fact.

To show Hypothesis (T3) holds for all surfaces we first note that if the
lengths of all closed geodesics were an integer multiple of some constant $d$,
then the length of every saddle connection would be an integer multiple of
$d/2$. To see this, let $s$ be any saddle connection on $X$. If $i(s)=t(s)$
then $s$ is a closed geodesic and so we are done. If $i(s)\neq t(s)$ then by
Proposition 6.9, there exists a closed saddle connection path $c_{i}$ such
that $c_{i}$ passes through $i(s)$ and that $\bar{s}c_{i}s$ forms a saddle
connection path (where $\bar{s}$ is the saddle connection $s$ with reversed
orientation). Similarly, there exists a closed saddle connection path $c_{t}$
which starts and ends at $t(s)$, such that $sc_{t}\bar{s}$ forms a saddle
connection path. Note that the concatenation $sc_{t}\bar{s}c_{i}$ is also a
closed saddle connection path of length $2\ell(s)+\ell(c_{t})+\ell(c_{i})$ and
so by Hypothesis (T3), $\ell(s)\in(d/2)\mathbb{N}$. Let us now assume for a
contradiction that (T3) does not hold and, in particular, the above property
holds for the saddle connection lengths.

Using results in [12], $X$ contains an embedded cylinder $C$ (the product of a
circle with an interval) whose boundaries consist of a single saddle
connection or multiple parallel saddle connections. We now aim to construct a
countable family of triangles using this cylinder (Figure 5).

$x$$x$$x$$x$$y$$y$$y$$b$$b$$b$$a$$T_{1}$$T_{2}$$T_{3}$$c_{3}$ Figure 5: Three
copies of a cylinder on $X$ with two singularities on separate boundaries
represented by circles and squares. The corresponding triangles $T_{1}$,
$T_{2}$ and $T_{3}$ are also drawn.

Fix two singularities $x$ and $y$, one from each boundary. Let $b$ denote the
union of saddle connections which form the boundary of the cylinder connecting
$x$ to itself. Let $a$ be one of the saddle connection connecting $x$ to $y$
across the cylinder such that the angle between $a$ and $b$ is acute. Then
consider the unique saddle connection $c_{n}$ connecting $x$ to $y$ which is
defined to be the third side in a triangle $T_{n}$ whose other edges are $b$
concatenated with itself $n$ times and a. By hypothesis each edge has length
which is an integer multiple of $d/2$. However, by elementary Euclidean
geometry we can show that this cannot hold for all sufficiently large $n$,
giving the required contradiction.

To derive an asymptotic estimate for $V(x,R)$ we can associate the complex
function

$\eta_{X}(z)=\int_{0}^{\infty}e^{-zR}dV(x,R).$

Let $\mathcal{P}(x):=\left\\{p\hbox{ : }i(p)=x\right\\}$ denote the set of
saddle connection paths starting at $x$. We can rewrite $\eta_{X}(z)$ as
follows:

$\displaystyle\eta_{X}(z)$
$\displaystyle=\int_{0}^{\infty}e^{-zR}\left(\frac{d}{dR}\hbox{\rm
Vol}(B(x,R))\right)dR$ $None$
$\displaystyle=2\pi(k(x)+1)\int_{0}^{\infty}e^{-zR}RdR+2\pi\sum_{p\in\mathcal{P}(x)}k(t(p))\int_{\ell(p)}^{\infty}e^{-zR}(R-\ell(p))dR$
$\displaystyle=2\pi(k(x)+1)\int_{0}^{\infty}e^{-zR}RdR+2\pi\sum_{p\in\mathcal{P}(x)}k(t(p))e^{-z\ell(p)}\int_{0}^{\infty}e^{-zR}RdR$
$\displaystyle=\frac{2\pi}{z^{2}}(k(x)+1)+\frac{2\pi}{z^{2}}\sum_{p\in\mathcal{P}(x)}k(t(p))e^{-z\ell(p)}$
$\displaystyle=\frac{2\pi}{z^{2}}(k(x)+1)+\frac{2\pi}{z^{2}}{\underline{v}}(z)\cdot(I-\widehat{M_{z}})^{-1}{\underline{u}},$

where $\underline{u}=(k(t(s_{j})))_{j=1}^{\infty}\in\ell^{\infty}(\mathbb{C})$
and
$\underline{v}(z)=(\chi_{\mathcal{E}_{x}}(s_{j})e^{-z\ell(s_{j})})_{j=1}^{\infty}\in\ell^{1}(\mathbb{C})$,
where $\chi_{\mathcal{E}_{x}}$ denotes the characteristic function of the set
$\mathcal{E}_{x}=\\{s\in\mathcal{S}\hbox{ : }i(s)=x\\}$ of saddle connections
starting from the singularity $x\in\Sigma$.

######  Lemma 6.10.

The function $\eta_{X}(z)$ is analytic for $Re(z)>h$ and has a meromorphic
extension to $Re(z)>0$. Moreover $\eta_{X}(z)$ has a simple pole at $z=h$ and
no other poles on $Re(z)=h$.

###### Proof.

We can apply the analysis of $(I-\widehat{M_{z}})^{-1}$ in Section 4 to (6.1),
where we use hypotheses (T1)-(T3) in place of (H1)-(H3). ∎

We can now apply Theorem 5.1 to deduce the following.

######  Theorem 6.11.

There exists a $C>0$ such that $V(x,R)\sim Ce^{hR}$ as $R\to+\infty$, i.e.,

$\lim_{R\to+\infty}\frac{V(x,R)}{e^{hR}}=C.$

Typically $C=C(x)$ will depend on the choice of $x$.

There is a closely related result for counting the number of geodesic arcs
$N_{X}(x,y,R)$ starting at $x\in\Sigma$ and finishing at $y\in\Sigma$.

######  Proposition 6.12.

There exists a $D>0$ such that $N_{X}(x,y,R)\sim De^{hR}$ as $R\to+\infty$,
i.e.,

$\lim_{R\to+\infty}\frac{N_{X}(x,y,R)}{e^{hR}}=D.$

###### Proof.

The proof simply requires replacing the function $\eta_{X}(z)$ by the function

$\eta_{N}(z)=\int_{0}^{\infty}e^{-zR}dN_{X}(x,R)={\underline{v}}(z)\cdot(I-\widehat{M_{z}})^{-1}{\underline{w}},$

where ${\underline{w}}=(\chi_{\mathcal{F}}(s_{i}))_{i=1}^{\infty}$, with
$\chi_{\mathcal{F}}(s)$ denoting the characteristic function for the set
$\mathcal{F}=\\{s\in\mathcal{S}\hbox{ : }t(s_{i})=y\\}$ of saddle connections
ending at the singularity $y$ and $u(z)$ was defined after equation (6.1).
Again the properties of $(I-\widehat{M_{z}})^{-1}$ allow one to apply Theorem
5.1 to deduce the result. ∎

######  Remark 6.13.

We conclude with some final remarks.

  1. 1.

It is not necessary for the ball in Theorem 6.11 to be centered at a
singularity. Let $y\in X-\Sigma$ and let $G$ be the set of geodesics $g$, from
$y$ to a singularity, such that $g$ has length $\ell(g)$. Order $G$ by non-
decreasing lengths. We define a matrix $P$ where

    1. (a)

the rows are indexed by such geodesics $g$ and the columns are indexed by the
oriented saddle connections $s$;

    2. (b)

the non-zero entries correspond to pairs $g$, $s$ such that:

      1. i.

The singularity $t(g)$ at the end of $g$ is the same as that $i(s)$ at the
start of the saddle connection $s$; and

      2. ii.

The geodesic $g$ and saddle connection $s$ have an angle of at least $\pi$
between them.

    3. (c)

The non-zero entries are $P_{z}(g,s)=e^{-z\ell(s)}$.

One can then modify the complex function to
$\eta_{X}(z)=\frac{2\pi}{z^{2}}(k(x)+1)+\frac{2\pi}{z^{2}}{\underline{v}_{p}(z)}\cdot\widehat{P_{z}}(I-\widehat{M_{z}})^{-1}\underline{u}$,
where $\underline{v}_{p}(z)=(e^{-z\ell(g)})_{g\in G}\in\ell^{1}(\mathbb{C})$
and then continue the proof as in Theorem 6.11.

  2. 2.

Theorem 6.11 also follows as a corollary of Theorem 6.12 by using a simple
approximation argument. In particular, this shows that
$C=D\int_{0}^{\infty}e^{-u}u^{2}du$.

  3. 3.

Let $L(x,R)$ be the total circumference of a circle centred at $x$ and whose
radius is a geodesic of length $R$. The same approach as in the proof of
Theorem 6.11 (or an approximation argument as in item 2 would give an
asymptotic formula of the form $L(x,R)\sim Ee^{hR}$, as $R\to+\infty$.

  4. 4.

Eskin and Rafi have announced a closely related asymptotic result to Theorem
6.11 for closed geodesics on $X$. By studying zeta functions $\zeta_{X}(z)$
instead of eta functions $\eta_{X}(z)$ they show that the number of closed
geodesics of length at most $R>0$ is asymptotic to $e^{hR}/hR$ as
$R\to+\infty$.

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