# Average diffusion rate of Ehrenfest Wind-tree billiards

Simon BARAZER

###### Abstract.

One of the versions of the wind-tree model of Boltzmann gas, suggested by Paul
and Tatiana Ehrenfest more than a century ago, can be seen as a billiard in
the plane endowed with $\mathbb{Z}\oplus\mathbb{Z}$-periodic rectangular
obstacles. In the breakthrough paper by V. Delecroix, P. Hubert and S.
Lelievre the authors proved, that the diffusion rate of trajectories in such a
billiard is equal to $\frac{2}{3}$, that is the maximal distance from the
origin achieved by a point of a typical trajectory on a segment of time
$[0,t]$ grows roughly as $t^{\frac{2}{3}}$ for large $t$. Here $\frac{2}{3}$
is the Lyapunov exponent of the associated renormalizing dynamical system.

This pioneering result does not tell, however, whether trajectories spend most
of the time close or far from the initial point. In the current paper, we
prove that the average distance from the origin grows with the same rate
$t^{\frac{2}{3}}$. In plain terms, it means that trajectories mostly stay as
far as possible from the initial point (though, it is known that the wind-tree
billiard is recurrent, so trajectories occasionally pass close to the initial
point).

More generally, fundamental rigidity results by A. Eskin and M. Mirzakhani
completed by certain genericity results by J. Chaika and A. Eskin imply that
the diffusion rate of almost all flat geodesic rays on any
$\mathbb{Z}^{d}$-cover of a closed translation surface $S$ is given by certain
Lyapunov exponent of the Kontsevich–Zorich cocycle on the
$\text{SL}_{2}(\mathbb{R})$-orbit closure of $S$. In this paper we prove that
in this most general setting, the average and maximum diffusion rates
coincide.

Figure 1. Trajectory in a Wind-Tree

##  1\. Introduction

The wind-tree model is an infinite $\mathbb{Z}^{2}$-periodic billiard in the
plane $\mathbb{R}^{2}$, where identical rectangular obstacles are placed in
such way that the corresponding sides are aligned with the horizontal and
vertical axes and the centers of rectangles are located at the lattice points,
see figure 1). More formally, for every couples of real parameters
$(a,b)\in(0,1)^{2}$ the wind-tree table $T(a,b)$ is defined as

$T(a,b)=\mathbb{R}^{2}\setminus\bigsqcup_{(m,n)\in\mathbb{Z}^{2}}\left(m-\frac{a}{2},m+\frac{a}{2}\right)\times\left(n-\frac{b}{2},n+\frac{b}{2}\right).$

It’s assumed that reflections on the sides of the rectangles are regular and
the speed of all trajectories is constant equal to one (figure 1). A version
of this model was introduced by Tatiana and Paul Erhenfest more than a century
ago, see [EE90]. It have been studied more recently by using the tools of
translation surfaces theory. Athur Avila and Pascal Hubert proved in [AH20]
that the trajectories in the wind-tree are recurrent. Later Vincent Delecroix,
Pascal Hubert and Samuel Lelièvre proved in [DHL14] that the diffusion rate of
a typical trajectory is equal to $\frac{2}{3}$. If $\gamma_{t}(x,\theta)$ is
the wind-tree flow in the direction $\theta$ starting from $x\in T(a,b)$, they
proved the following theorem.

######  Theorem (Delcroix, Hubert, Lelievre, 2014).

For any $(a,b)\in(0,1)^{2}$, for almost every $\theta$ (depending on $(a,b)$)
and for all $x\in T(a,b)$ having an infinite future orbit, the following
relation holds:

$\limsup_{t\rightarrow+\infty}\frac{\log d(x,\gamma_{t}(x,\theta))}{\log
t}=\frac{2}{3}\,.$

Anton Zorich and Vincent Delecroix generalized this result to a large class of
wind-tree billiards with more complicated obstacles [DZ20]. Charles Fougeron
studied in [Fou20] orbit closures of flat surfaces associated to even more
general wind-tree billiards. In particular, he identified the Lyapunov
exponent of the Kontsevich–Zorich cocycle that governs the associated
diffusion rate. In a different direction, Krzysztof Fraczek and Corinna
Ulcigrai showed in [FU14] that the directional flow for almost all wind-tree
billiard is non-ergodic.

Note that the diffusion rate $\frac{2}{3}$ is defined in the original paper
[DHL14] of Delecroix–Hubert–Lelièvre as the maximal distance from the origin
achieved by a point of a typical trajectory on a large segment of time. In the
present paper, we study the diffusion in average. In particular, we prove the
following theorem:

######  Theorem 1.

For any $(a,b)\in(0,1)^{2}$, for almost every $\theta$ (depending on $(a,b)$)
and for all $x\in T(a,b)$ having an infinite future orbit, the following
relation holds:

$\lim_{T\rightarrow+\infty}\frac{\log\frac{1}{T}\int_{0}^{T}d(x,\gamma_{t}(x,\theta))dt}{\log
T}=\frac{2}{3}\,.$

The strategy of the proof is similar to the proof of the Theorem of
Delecroix–Hubert–Lelièvre presented above. The first step is standard:
following the Katok–Zemliakov construction ([ZK75], see also [MT02]), the
authors of [DHL14] construct a translation surface $X_{\infty}(a,b)$ without
boundary from four copies of the wind-tree table $T(a,b)$. By construction,
the billiard flow in $T(a,b)$ is conjugate to the geodesic flow on the
translation surface $X_{\infty}(a,b)$. The fact that the wind-tree table is
$\mathbb{Z}^{2}$-periodic implies that the surface $X_{\infty}(a,b)$ is a
$\mathbb{Z}^{2}$-cover over a compact translation surface $X(a,b)$. In this
way, we reduce the problem to a particular case of the study of geodesics on
$\mathbb{Z}^{d}$-covers of translation surfaces.

We call $\tilde{S}\to S$ a lattice cover if $\tilde{S}$ is a connected
$\mathbb{Z}^{d}$-cover of a compact translation surface $S$. We will see that
such a cover is defined by a cohomology class $f$ in the first cohomology
$H^{1}(S,\mathbb{Z}^{d})$. As in [DHL14], the diffusion of a billiard
trajectory can be studied through the intersection pairing of $f$ with
geodesic segments on $S$ completed to closed paths in a natural way. We denote
by $\tilde{\phi}_{t}(x,\theta),\phi_{t}(x,\theta)$ the geodesic flows on
$\tilde{S}$ and $S$ respectively. By $d(x,y)$ we denote the distance between
points $x$ and $y$ on the translation surface $\tilde{S}$. We prove the
following theorem.

######  Theorem 2.

Let $\tilde{S}\rightarrow S$ a lattice cover; let $f$ be the associated
cohomology class in $H^{1}(S,\mathbb{Z}^{d})$. For every $x\in\tilde{S}$ and
almost every $\theta$ we have

$\lim_{T\rightarrow+\infty}\frac{\log\frac{1}{T}\int_{0}^{T}d(x,\tilde{\phi}_{t}(x,\theta))dt}{\log
T}=\Lambda(f),$

where $\Lambda(f)$ is the Lyapunov exponent of $f$ for the Kontsevich–Zorich
cocycle and with respect to the affine invariant measure supported by the
orbit closure of $S$ for the action of $\text{SL}_{2}(\mathbb{R})$.

The proof of this proposition uses tools from dynamics on the moduli spaces of
translation surfaces. The corresponding technique was developed by A. Zorich
in [Zor99], by G. Forni in [FM14] and applied in [DHL14]. Let
$\pi:\tilde{S}\to S$ be a lattice cover, and let $f$ be the associated
cohomology class. Let $(\tilde{x},x)$ be a pair of points in $\tilde{S}$ and
in $S$ respectively, such that $\pi(\tilde{x})=x$. To study the drift of a
trajectory starting from $\tilde{x}$ we project the trajectory to the base
$S$. We consider a small horizontal interval $I$ on $S$ with one of the
endpoints at $x$ and the first return map of the vertical geodesic flow to
$I$. The first return map is generally an interval exchange transformation;
the vertical flow on $S$ can be represented by Veech’s zippered rectangles
construction as a suspension flow over this interval exchange transformation.
Consider a piece of trajectory starting at $\tilde{x}$ and going up to the
$N$th visit to the interval $I$. Joining the endpoints of such piece of
trajectory along $I$, we get a cycle $C_{N}(S,x)$ in the first homology of
$S$. The drift of the trajectory in $\tilde{S}$ is given by the intersection
pairing of this cycle with the cohomology class $f$:

(1.1) $\langle f,C_{N}(S,x)\rangle\,.$

Following Zorich and Forni, we use a renormalization procedure given by the
Teichmüller flow on the moduli space of translation surfaces to study the
behavior of the above quantity for large $N$. The Teichmüller flow $g_{s}$
contracts vertical trajectories, so for an appropriate choice of time $s(N)$
cycles $C_{N}(S,x)$ representing long trajectories of $\phi_{t}$ (and thus
having large norm) are mapped to cycles on the new surface $g_{\tau}\cdot S$
having norm comparable to $1$. The growth of expression (1.1) is then given by
the Kontsevitch–Zorich cocycle governed by the Gauss–Manin connection on the
Hodge bundle. If we fix a cross section for the Teichmüller flow in the moduli
space of translation surfaces, we obtain a sequence of flat surface $S_{n}$ in
the cross section such that

$g_{s_{n}}\cdot S=\Gamma_{n}\cdot S_{n}\,,$

where $\Gamma_{n}$ is an element of the mapping class group. The behavior of
the expression (1.1) for large $N$ can be related to the behavior of the
sequence $\Gamma_{n}^{*}f$, which is given by the Oseledets theorem.

In Section 2 we recall some classical results in the theory of translation
surfaces. Namely, we recall the definition of moduli spaces of translation
surfaces, the action of $\text{SL}_{2}(\mathbb{R})$ and we define the
Kontsevich–Zorich cocycle on the Hodge bundle. We also recall fundamental
theorems on dynamics on moduli spaces of flat surfaces.

In Section 3 we prove Theorem 2. Using Veech’s zippered rectangles
construction presented in Section 3, we relate the statement of the Theorem to
the behavior of the sum

$\sum_{k\leq\textsc{n}}\langle f,C_{k}(S,x)\rangle$

for large n. We define a convenient cross-section for the Teichmüller flow in
Section 3. In Section 3 we use this cross section and the renormalization
procedure to construct trajectories with a good excursion rates by applying
the Oseldet’s theorem. In Section 3 we establish the upper bound in Theorem 2
by using the Oseldet’s theorem. To obtain the lower bound we use the
trajectories constructed in Section 3 and a uniform upper bound to control the
diffusion near these excursions. Finally, a proof of a technical Lemma 3 is
placed to an Appendix.

### Acknowledgements.

I am very grateful to Pascal Hubert for supporting me during this work at the
I2M of Marseille, and introducing me to Teichmüller dynamics and its
applications. I am also very grateful to Anton Zorich for a careful read of
this paper.

##  2\. Background facts and constructions

The present paper uses tools from the theory of compact flat surfaces. We
refer to the surveys of A. Zorich [Zor06] and of J.C. Yoccoz [Yoc07] for an
elementary introduction to the subject and a survey [FM14] of G. Forni and C.
Matheus for more details. In this section, we introduce the material which
will be used in the proof of Theorems 1 and 2 announced in the introduction.
We start by presenting some basic facts on flat surfaces and lattice
coverings. We also define the Teichmüller and moduli spaces of flat surfaces
and give recent results related to the dynamics of the action of
$\text{SL}_{2}(\mathbb{R})$. Finally we recall Veech’s zippered rectangle
construction which we use in the study of the geodesic flow of a flat surface.

### Compact flat surfaces

A compact translation surface $S$ is commonly defined as a pair $(S,\omega)$
where $S$ is a compact Riemann surface and $\omega$ is a non-zero holomorphic
differential on $S$. The one-form $\omega$ defines a flat metric
$ds^{2}=\text{Re}(\omega\otimes\bar{\omega})$. Compactness of the surface $S$
implies that the induced area of $S$ is necessarily finite.

The metric as above has singularities at the finite set $\Sigma_{\omega}$ of
zeros of $\omega$; at a point $x\in\Sigma_{\omega}$ the metric has a conical
singularity of angle $2\pi(k_{x}+1)$, where $k_{x}$ is the order of the zero.
If $g$ is the genus of the surface, the Gauss–Bonnet formula gives

$\sum_{x}k_{x}=2g-2\,.$

By taking local primitives of $\omega$ in simply-connected domains in the
complement to $\Sigma_{\omega}$, we get an atlas of charts on
$S\backslash\Sigma_{\omega}$ with values in $\mathbb{C}$ and such that the
transition functions are translations.

The geodesic flow is defined on the tangent bundle to
$S\backslash\Sigma_{\omega}$. A trajectory may land to a singularity in a
finite time in onward or in a backward direction. Such trajectories are called
singular trajectories.

The translation structure allows to trivialize the tangent bundle over
$S\backslash\Sigma_{\omega}$. Thus, a geodesic on a translation surface
preserves the initial direction, which allows to consider a directional flow
in any fixed direction $\theta$. We mainly consider the vertical (pointing to
the North) and the horizontal (pointing to the Est) directional flows. The
geodesic flow on a flat surface has been studied extensively in the past
decades. The following fundamental result concerning the directional flows is
due to S. Kerckhoff, H. Masur and P. Smillie [KMS86]

######  Theorem (Kerckoff, Masur, Smillie).

For every translation surface $S$, the directional flow in direction $\theta$
is uniquely ergodic for almost all directions $\theta$.

A saddle connection on a flat surface is a singular trajectory which connects
two singularities (possibly the same). We are particularly interested in the
vertical and horizontal saddle connections. The following result from [Kea75]
is often called Keane’s criterion.

######  Theorem (Keane [Kea75]).

If a flat surface has no saddle connections in direction $\theta$, then the
geodesic flow in this direction is minimal.

### Lattice coverings

A lattice cover of a translation surface is a triple $(\tilde{S},S,\pi)$.
Where $S$ is a compact translation surface, $\tilde{S}$ is a connected
translation surface, and $\pi:\tilde{S}\rightarrow S$ is a holomorphic
$\mathbb{Z}^{d}$-covering map. Moreover, we assume that the holomorphic one-
form $\tilde{\omega}$, defining the translation structure on $\tilde{S}$, is
the pullback of the holomorphic one-form $\omega$ defining the translation
structure on $S$. That is $\pi^{*}\omega=\tilde{\omega}$. Lattice
$\mathbb{Z}$-covers were studied in [HW12] and [HHW13]. For instance, we can
think to a surface embedded in an highter dimentionnal torus and periodic in
d-directions.

By definition, the monodromy group of a lattice cover is isomorphic to
$\mathbb{Z}^{d}$. We denote his action on $\tilde{S}$ by $\mathbf{n}\cdot x$.
If we fix a base point $x$ on $S$, an isomorphism of the monodromy group with
$\mathbb{Z}^{d}$ defines a morphism

$f:\pi_{1}(S,x)\longrightarrow\mathbb{Z}^{d}.$

This morphism factorizes thought the Abelianised group and defines a class $f$
in the first cohomology of $S$:

$f\in\text{Hom}(H_{1}(S,\mathbb{Z}),\mathbb{Z}^{d})=H^{1}(S,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Z}^{d}.$

The class $f$ characterizes the cover up to an isomorphism. We will use the
cohomology class $f$ to measure the deviation of a trajectory.

### Teichmüller spaces

The Teichmüller spaces of flat surfaces are spaces of isotopy classes of flat
surfaces with prescribed singularities (we refer to [Yoc07],[FM14] and [Zor06]
for an introduction to the subject). We fix a compact topological surface $M$
of genus $g$ and a set $\Sigma=\\{x_{1},...,x_{r}\\}$ of distinct points in
$M$. Let $\kappa=(\kappa_{1},...,\kappa_{r})$ a set of non negative integers
such that

$\sum_{l=1}^{r}\kappa_{l}=2g-2,$

The Teichmüller space $\mathcal{TH}(\kappa)$ associated to $(M,\Sigma,\kappa)$
is the set of isotopy classes of compact marked translation surfaces
$\phi:S\rightarrow M$ such that, for all $i$ the surface $S$ have a conical
singularity of angle $2\pi(\kappa_{i}+1)$ at $\phi^{-1}(x_{i})$ (a zeros of
order $\kappa_{i}$) and no singularities elsewhere.
The Teichmüller spaces have a nice structure of complex linear manifolds, the
local charts are given by taking period coordinates in the relative cohomology

$(\phi,S)\rightarrow[\phi_{*}\omega]\in H^{1}(M,\Sigma,\mathbb{C}).$

The period map is a local homeomorphism (see [Yoc07]).

The mapping class group $\text{Mod}(M,\Sigma)$ is the group of isotopy classes
of diffeomorphisms on $(M,\Sigma)$, it acts naturally on the Teichmüller
spaces. The moduli space $\mathcal{H}(\kappa)$ is the quotient space of
$\mathcal{TH}(\kappa)$ by $\text{Mod}(M,\Sigma)$. Flat surfaces can have
nontrivial automorphism groups; $\mathcal{H}(\kappa)$ is a complex orbifold in
general, but we can find a finite cover which is a manifold. The area of a
surface define a function on the moduli and Teichmüller space and we denote
$\mathcal{TH}^{1}(\kappa),\mathcal{H}^{1}(\kappa)$ the level set of surfaces
with an area equal to one.

######  Remark 1 (Pointed surfaces).

We can add marked points that are regular to surfaces in Teichmüller spaces;
regular means that they do not correspond to conical singularities of the
underlying surface. We call such surfaces: pointed surfaces. For a given
$\kappa$ we denote by
$\widetilde{\mathcal{TH}}(\kappa)=\mathcal{TH}((0,\kappa))$ the Teichmüller
space of flat surfaces in $\mathcal{TH}(\kappa)$ with a marked regular point
(it’s the universal curve over $\mathcal{TH}(\kappa)$). There is a natural
projection $\widetilde{\mathcal{TH}}(\kappa)\to\mathcal{TH}(\kappa)$ that is
invariant under the action of the mapping class group. An automorphism of a
flat surface is locally given by a translation, if an automorphism stabilise a
regular point then it’s locally constant near this point and then it’s
globally trivial when the surface is connected. Using this we see that the
moduli spaces $\widetilde{\mathcal{H}}(\kappa)$ of pointed surfaces are
manifolds.

### Dynamics in the moduli space of translation surfaces

In this paper we use the left action of $\text{SL}_{2}(\mathbb{R})$ on the
moduli spaces of translation surfaces $\mathcal{H}(\kappa)$ (see [Zor06] or
[FM14] for a survey). The group acts by linear deformations of the flat
structure and preserves the area of flat surfaces then it also preserves the
hypersuface $\mathcal{H}^{1}(\kappa)$. The action of the group
$\text{SL}_{2}(\mathbb{R})$ in period coordinates can be seen as the diagonal
action on the coefficients $\mathbb{C}\simeq\mathbb{R}^{2}$. Two one-parameter
subgroups of $\text{SL}_{2}(\mathbb{R})$ are important in the study of
translation surfaces: the diagonal subgroup and the subgroup
$\text{SO}_{2}(\mathbb{R})$. The flow induced by the diagonal subgroup is
denoted by

(2.1) $g_{s}=\begin{pmatrix}e^{s}&0\\\ 0&e^{-s}\end{pmatrix}$

and called the Teichmüller flow. This flow shrinks the flat metric in the
vertical direction and dilates the metric in the horizontal direction. The
flow $r_{\theta}$ is defined by the action of $\text{SO}_{2}(\mathbb{R})$:

$r_{\theta}=\left(\begin{array}[]{rr}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{array}\right).$

It does not change the flat metric, but it changes the distinguished direction
(“direction of the North”).

The vector space $H^{1}(S,\Sigma;\mathbb{C})$ serving for period coordinates
admits a natural integer lattice $H^{1}(S,\Sigma;\mathbb{Z}\oplus
i\mathbb{Z})$, which endows it with a canonical volume element: the one, for
which a fundamental domain of the lattice has a unit volume. This volume
element defined in period coordinates induces a well-defined volume element in
every stratum $\mathcal{H}(\kappa)$ of translation surfaces called the
Masur–Veech volume element. By construction the group
$\text{SL}_{2}(\mathbb{R})$ preserves the Masur–Veech volume element and the
flat area of the surfaces. Thus, the Masur–Veech volume element in
$\mathcal{H}(\kappa)$ induces a natural volume element on the hypersurface
$\mathcal{H}^{1}(\kappa)$. The following Theorem proved independently by H.
Masur [Mas82] and by W. Veech [Vee78] is the keystone result in dynamics in
the moduli space of translation surfaces.

######  Theorem (Masur, Veech).

The total volume of any stratum $\mathcal{H}^{1}(\kappa)$ in the moduli space
of compact translation surfaces is finite. The Teichmüller flow acts
ergodically on every connected component of every stratum.

### Orbit closures and affine invariant submanifolds

Let $S$ be a translation surface of area $1$ in some stratum
$\mathcal{H}(\kappa)$, $S\in\mathcal{H}^{1}(\kappa)$. Detecting the orbit
closure $\mathcal{H}^{1}_{S}=\overline{\text{SL}_{2}(\mathbb{R})\cdot S}$
(resp $\mathcal{H}_{S}=\overline{\text{GL}_{2}(\mathbb{R})\cdot S}$) of the
orbit of $S$ in $\mathcal{H}^{1}(\kappa)$ is one of the most important
questions in the theory of translation surfaces. The Magic Wand theorem of A.
Eskin–M. Mirzakhani–A. Mohammadi [eskin2013invariant], [EMM15] (and an
important complement by S. Filip [Fil16]) provide the general structure of
$\mathcal{H}^{1}_{S}$.

######  Theorem (Eskin–Mirzakhani–Mohammadi).

For each translation surface $S$ the orbit closure $\mathcal{H}_{S}$ is
locally a linear subspace in period coordinates (possibly a finite collection
of linear subspaces).

The orbit closure $\mathcal{H}_{S}$ admits a measure $\tilde{\mu}_{S}$ such
that

  * •

Locally, the measure $\tilde{\mu}_{S}$ is proportional to the Lebesgue measure
in the period coordinates.

  * •

The measure $\mu_{S}$ induced on $\mathcal{H}^{1}_{S}$ is an invariant
probability measure for the Teichmüller flow.

  * •

The measure $\mu_{S}$ is ergodic for the Teichmüller flow.

### Hodge bundle and Kontsevich–Zorich cocycle

Every stratum $\mathcal{H}(\kappa)$ admits two natural vector bundles
$\hat{H}^{1}\mathcal{H}(\kappa)$ and $H^{1}\mathcal{H}(\kappa)$ with fibers
$H^{1}(S,\Sigma_{S},\mathbb{R})$ and $H^{1}(S,\mathbb{R})$ over a point in
$\mathcal{H}(\kappa)$ represented by a translation surface $S$. Both bundles
carry a natural flat connection called the Gauss–Manin connection uniquely
determined by the property of equivariance of lattices
$H^{1}(S,\Sigma_{S},\mathbb{Z})$ and $H^{1}(S,\mathbb{Z})$ respectively. The
Kontevich–Zorich cocycles correspond to the monodrony of these connections
along the $\text{SL}_{2}(\mathbb{R})$-action. These cocycles, which we denote
by $G_{s}$, serve for an efficient renormalization procedure.

The bundle $H^{1}\mathcal{H}(\kappa)$ is endowed with a natural norm
$\|.\|_{h}$ called the Hodge norm; this norm is induced from the classical
Hodge norm on $H^{1,0}(S,\mathbb{C})$ by the canonical isomormhism

$H^{1}(S,\mathbb{R})\longrightarrow H^{1,0}(S,\mathbb{C})\,.$

G. Forni proved in [FM14] the following uniform upper bound for the growth
rate of the Hodge norm along the Teichmüller flow:

######  Theorem (Forni, [FM14]).

For every $u\in H^{1}\mathcal{H}(\kappa)$ such that $\|u\|_{h}=1$ the
following bound holds:

$\frac{d\|G_{s}u\|_{h}}{ds}\Big{|}_{s=0}\leq 1\,.$

This bound implies, in particular, integrability of the Kontsevich–Zorich
cocycle for any affine invariant probability measure on
$\mathcal{H}^{1}(\kappa)$.

### Oseledets and Birkhoff ergodic theorems

A. Eskin and J. Chaika have proved in [CE15] the following important versions
of the Birkhoff and Oseldets ergodic theorems for the measure $\mu_{S}$ and
the Kontsevich–Zorich cocycle. These theorems are particularly efficient in
the study of translations surfaces (see also the survey of J. Chaika [CE15]).

######  Theorem (Chaika–Eskin).

For any translation surface $S$ in any stratum $\mathcal{H}^{1}(\kappa)$, for
almost any angle $\theta$ and for any continuous function $\phi$ with compact
support in $\mathcal{H}^{1}_{S}$ the following equality holds:

(2.2)
$\displaystyle\lim_{T\rightarrow+\infty}\frac{1}{T}\int_{0}^{T}\phi(g_{s}r_{\theta}S)ds=\int_{\mathcal{H}^{1}(S)}\phi~{}d\mu_{S}\,.$

######  Theorem (Chaika–Eskin).

For any translation surface $S$ in any stratum $\mathcal{H}^{1}(\kappa)$, and
for almost any angle $\theta$, the translation surface $r_{\theta}S$ is
Oseledets-generic for the Kontsevich–Zorich cocycle along the Teichmüller flow
with respect to the measure $\mu_{S}$.

######  Remark 2 (cocycles with multiple components).

Recall that a cohomology class $f\in
H^{1}(S,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Z}^{d}$ associated to a
$\mathbb{Z}^{d}$-lattice cover has $d$ components $f=(f_{i})_{1\leq i\leq d}$
and takes values in $\mathbb{Z}^{d}$. In what follows the Hodge-norm of such a
cohomology class is defined as the sup-Hodge norm of components
$\|f\|_{h}:=\max_{i}\|f_{i}\|_{h}$. Thus, the Lyaponov exponent of $f$ with
respect to the Kontsevich–Zorich cocycle the along the Teichmüller flow is the
maximum of the Lyapunov exponents of the components
$\Lambda(f)=\max_{i}\Lambda(f_{i})$.

### Zippered rectangles construction

Veech’s zippered rectangles construction is a particularly convenient way to
represent a vertical flow on a compact translation surface, see [Vee78] for
the original definition or [Yoc07] for a survey. We recall briefly an idea of
the construction. Consider a horizontal segment $I$ on a translation surface
$S$. We always assume that $I$ does not have conical singularities inside it
(a conical singularity at one of the extremities of $I$ is allowed and
sometimes even assumed). The first return map $T:I\to I$ of the vertical flow
on $S$ to $I$ is an interval exchange transformation: it chops the interval
$I$ into several subintervals and rearranges them in a different order,
preserving the orientation and avoiding overlaps (see the surveys [Yoc07] and
[FM14], [Via08] for details on interval exchange transformations). More
formally, we have two decompositions of $I$:
$I^{t}=\sqcup_{\alpha}I_{\alpha}^{t},~{}~{}~{}~{}I^{b}=\sqcup_{\alpha}I_{\alpha}^{b}$
indexed by a set $\mathcal{A}$; here $I\backslash I^{\cdot}$ is a finite set
of singularities. The map $T$ sends $I_{\alpha}^{t}$ to $I_{\alpha}^{b}$ by a
translation; we generally denote $\lambda_{\alpha}$ the length of
$I_{\alpha}^{\cdot}$. We denote $I^{*}$ the set of points with an infinite
future orbit; the map $T:I^{*}\to I^{*}$ is well defined. The return time of
the vertical flow is piecewise constant on $I$ and constant on each of
subintervals $I_{\alpha}^{t}$, we denote it $\tau_{\alpha}$. Thus, a choice of
the horizontal subinterval $I$ as above defines a decomposition of the
translation surface into a collection of rectangles — flow boxes
$I_{\alpha}^{t}\times[0,\tau_{\alpha}[$ of the vertical flow. This
decomposition changes, of course, when we change the base interval $I$. We
will mostly assume that $I$ has unit length and that the translation surface
$S$ has unit area.

### Coding the vertical geodesic flow

Let $S$ be a translation surface, $I$ be a horizontal segment of length one on
$S$ as above and $T:I\to I$ the first return map of the vertical flow to $I$.
We assume that the left endpoint $x_{0}$ of $I$ is nonsingular. By
$S^{\prime}$ we denote a pair $S^{\prime}=(S,x_{0})$. For any positive integer
$n$ and $x\in I$ we construct $n$-th return cycle
$C_{\textsc{n}}(S^{\prime},x)$ as follows. Consider an oriented segment
$\gamma_{\textsc{n}}(x)$ of the trajectory of the vertical flow starting from
$x$ and going up to $T^{\textsc{n}}(x)$. Completing the resulting path to a
closed path by a subsegment of $I$ starting from $T^{\textsc{n}}(x)$ and
ending at $x$ we get a cycle

$C_{\textsc{n}}(S^{\prime},x)\in H_{1}(S,\mathbb{Z}),$

with $S^{\prime}=(S,x_{0})$ (more precisely, the cycles can be lifted to the
homology of $S\backslash\Sigma_{S^{\prime}}$ with $\Sigma_{S}$ the set of
singularities). This is an additive cocycle for the interval exchange map $T$,
it satisfies

$C_{\textsc{n}+1}(S^{\prime},x)=C_{\textsc{n}}(S^{\prime},x)+C_{1}(S^{\prime},T^{\textsc{n}}x).$

We also denote $C_{\textsc{n}}(S^{\prime})$ the return cycle of the trajectory
starting from $x_{0}$, it satisfies

$C_{\textsc{n}}(S^{\prime})=\underset{x\rightarrow
x_{0}}{\lim}C_{\textsc{n}}(S^{\prime},x).$

Note that for all interior points $x\in I_{\alpha}^{t}$ the first return
cycles $C_{1}(S^{\prime},x)$ are homologous. We denote by $h_{\alpha}$ the
resulting first return cycles. When the vertical flow is minimal, these cycles
form a basis of $H_{1}(S\backslash\Sigma_{S^{\prime}},\mathbb{Z})$.

##  3\. Large excursions

In this section, we prove the main theorems of the paper. We first show
briefly how we can reduce the problem from lattice coverings of translation
surfaces to compact translation surfaces. This was previously discussed in the
paper [DHL14]. In this same section, we rephrase the theorem 2 by using the
formalism of zippered rectangle construction 2. In a second, by using Veech
zippered rectangles construction, we choose a convenient cross section for the
Teichmüller flow and then consider the first return map on this section. To
define the cross section, we use the lemma 3 which is given in the appendix.
In the third section, by using the normalization procedure, we produce a
sequence of trajectories with a good diffusion rate, and we show that these
excursions occur at exponential times. In fourth, from the results of A.
Zorich [Zor97], we obtain an upper bound for the diffusion. We use this upper
bound to control the trajectories around a large excurssion and also to state
the upper bound in theorem 2. Finally, we prove the lower bound, which is the
most tricky part. By using the excursions, we can show that the trajectory
goes far from the initial point; with the upper bound, we can control the
diffusion near these excursions and show that a typical trajectory spends a
large part of its time far from the starting point.

### Preliminary results on coverings

Let $\pi:\tilde{S}\rightarrow S$ be a lattice covering of a translation
surface $S$ with cocycle $f$. The flat structure defines a metric on
$\tilde{S}$, and we denote $d_{\tilde{S}}$ the flat distance on $\tilde{S}$.
Let $x,y$ be two points in the same fiber of $\pi$. Using the $\mathbb{Z}^{d}$
action, there is a unique $\mathbf{n}\in\mathbb{Z}^{d}$ such that
$y=\mathbf{n}\cdot x$. The lemma below allows us to compare the geodesic
distance between $x,y$ and the norm of $\mathbf{n}$ (for instance,
$\|\mathbf{n}\|=\sum_{i}|\mathbf{n}_{i}|$).

######  Lemma 1.

For all lattice covering $\tilde{S}\to S$, with a choice of cocycle $f$. There
exists a constant $A_{0}>0$ such that: for every $\tilde{x}\in\tilde{S}$ and
$\mathbf{n}\in\mathbb{Z}^{d}$, we have

$A_{0}^{-1}\|\mathbf{n}\|\leq d_{\tilde{S}}(x,\mathbf{n}\cdot x)\leq
A_{0}\|\mathbf{n}\|.$

Using this lemma, we can approach the deviation of the geodesic flow on
$\tilde{S}$. To do this, we use Veech zippered rectangle construction for the
surface $S$. Let $\tilde{x}\in\tilde{S}$, we denote
$\tilde{\phi}_{t}(\tilde{x})$ the geodesic flow on $\tilde{S}$ in the vertical
direction. Let $x=\pi(\tilde{x})$ be the projection on $S$. We assume that $x$
has an infinite future orbit, and $S$ is constructible by zippered rectangles
with the interval $I$ of length $1$ starting from $x$ and going to the east.
As before, we denote $S^{\prime}=(S,x)$ and $\phi_{t}(x)$ the vertical flow on
$S$. Let $(t_{\textsc{n}})_{\textsc{n}\geq 0}$ be the sequence of return times
of $(\phi_{t}(x))_{t\geq 0}$, on the interval $I$, and
$C_{\textsc{n}}(S^{\prime})$ be the $\textsc{n}-$th return cycle of the
trajectory starting from $x$.

######  Lemma 2.

Under these assumptions, there exist $A_{1},B_{1}$ such that, for all
$\textsc{n}>0$

$\frac{A_{1}^{-1}}{\textsc{n}}\underset{k\leq\textsc{n}}{\sum}\|\langle
f,C_{k}\rangle\|-B_{1}\leq\frac{1}{T}\int_{0}^{T}d(\tilde{x},\tilde{\phi}_{t}(\tilde{x}))dt\leq\frac{A_{1}}{\textsc{n}}\underset{k\leq\textsc{n}}{\sum}\|\langle
f,C_{k}\rangle\|+B_{1},~{}~{}~{}\forall
T\in[t_{\textsc{n}},t_{\textsc{n}+1}].$

###### Proof.

Let $\mathbf{n}\in\mathbb{Z}^{d}$ and let $\gamma$ be any arc connecting $x$
and $\mathbf{n}\cdot x$. The vector $\langle f,\pi_{*}\gamma\rangle$ is then
equal to $\mathbf{n}$. For any $\textsc{n}\in\mathbb{N}$, the point
$\tilde{\phi}_{t_{\textsc{n}}}(\tilde{x})$ belongs to a unique leaf of $I$. If
$\tilde{I}$ is the leaf starting from $\tilde{x}$, then
$\tilde{\phi}_{t_{\textsc{n}}}(\tilde{x})$ belongs to
$\mathbf{n}\cdot\tilde{I}$, where $\mathbf{n}=\langle
f,C_{\textsc{n}}\rangle$. By using the triangular inequality, we can obtain
the following inequality:

$A_{0}^{-1}\|\langle f,C_{\textsc{n}}\rangle\|-1\leq
d(x,\tilde{\phi}_{t_{\textsc{n}}}(\tilde{x}))\leq A_{0}\|\langle
f,C_{\textsc{n}}\rangle\|+1,$

where $1$ is the length of the segment $I$. Using triangular inequality again,
we can write

$|d(x,\tilde{\phi}_{t_{\textsc{n}}}(\tilde{x}))-d(x,\tilde{\phi}_{t}(\tilde{x}))|\leq
d(\tilde{\phi}_{t_{\textsc{n}}}(\tilde{x}),\tilde{\phi}_{t}(\tilde{x}))\leq|t-t_{\textsc{n}}|.$

The last inequality uses the fact that the velocity is constant and equal to
one. The increments $(t_{k+1}-t_{k})_{k\leq\textsc{n}}$ take their values in a
finite set of strictly positive numbers (because in the Veech zippered
rectangle construction, the return time is constant on each sub-interval of
$I$). Then there is $B_{0}>0$ with

$B_{0}^{-1}\leq t_{k+1}-t_{k}\leq B_{0},~{}~{}~{}~{}\forall k\geq 0,$

and this implies

$\textsc{n}B_{0}^{-1}\leq T\leq B_{0}(\textsc{n}+1),~{}~{}~{}~{}\forall
T\in[t_{\textsc{n}},t_{\textsc{n}+1}[.$

Using these inequalities, we can estimate the integral by splitting the
interval $[0,T]$ into subintervals $[t_{k},t_{k+1}[$ for $k<\textsc{n}$ and
$[t_{\textsc{n}},T]$. We have

$\displaystyle\frac{1}{T}\int_{0}^{T}d(\tilde{x},\tilde{\phi}_{t}(\tilde{x}))dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{T}\sum_{k<\textsc{n}}\int_{t_{k}}^{t_{k+1}}d(\tilde{x},\tilde{\phi}_{t}(\tilde{x}))dt~{}~{}+~{}~{}\frac{1}{T}\int_{t_{\textsc{n}}}^{T}d(\tilde{x},\tilde{\phi}_{t}(\tilde{x}))dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{T}\sum_{k<\textsc{n}}\int_{t_{k}}^{t_{k+1}}d(\tilde{x},\tilde{\phi}_{t_{k}}(\tilde{x}))+(t-t_{k})dt~{}~{}+~{}~{}\frac{1}{T}\int_{t_{\textsc{n}}}^{T}d(\tilde{x},\tilde{\phi}_{t}(\tilde{x}))dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{T}\sum_{k<\textsc{n}}(t_{k+1}-t_{k})(A_{0}\|\langle
f,C_{k}\rangle\|+(t_{k+1}-t_{k}))~{}+~{}t_{\textsc{n}+1}-t_{\textsc{n}}$
$\displaystyle\leq$
$\displaystyle\frac{A_{0}B_{0}}{T}\underset{k<\textsc{n}}{\sum}\|\langle
f,C_{k}\rangle\|+\frac{\textsc{n}B_{0}^{2}}{T}+B_{0}$ $\displaystyle\leq$
$\displaystyle\frac{A_{0}B_{0}^{2}}{\textsc{n}}\underset{k<\textsc{n}}{\sum}\|\langle
f,C_{k}\rangle\|+B_{0}^{3}+B_{0}$

And this implies the existence of two constants $A_{1},B_{1}$ such that, for
all n we have

$\frac{1}{T}\int_{0}^{T}d(x,\tilde{\phi}_{t}(\tilde{x},\theta))dt\leq\frac{A_{1}}{\textsc{n}}\underset{k\leq\textsc{n}}{\sum}\|\langle
f,C_{k}(S^{\prime})\rangle\|+B_{1},~{}~{}~{}\forall
T\in[t_{\textsc{n}},t_{\textsc{n}+1}[.$

Similarly, we can obtain the lower bound and prove the lemma 2. ∎

If we are under the assumption of the last lemma and if $\Lambda f>0$. To
prove the theorem 2, it’s sufficient to prove the following: for all
$\epsilon>0$, it exists $\textsc{n}_{0}$ such that

(3.1) $\exp((\Lambda
f+1-\epsilon)\log(n))\leq\underset{k\leq\textsc{n}}{\sum}\|\langle
f,C_{t_{k}}\rangle\|\leq\exp((\Lambda
f+1+\epsilon)\log(\textsc{n})),~{}~{}~{}~{}\forall\textsc{n}\geq\textsc{n}_{0}.$

We will prove this statement in the rest of the paper. The following remark in
important.

######  Remark 3 (Restriction to the case $d=1$).

By using remark 2 restrict to cocycle $f$ with values in $\mathbb{Z}$, if
$f=(f_{i})_{i=1...d}$, the Lyapunov exponent is $\max_{i}\Lambda f_{i}$ and
also for the diffusion rate.

### Cross section for the Teichmüller flow

We fix a pointed surface $S^{\prime}=(S,x_{0})$ of area one and let
$\mu_{S^{\prime}},\mu_{S}$ be the two affine measures on the orbits
$\mathcal{H}^{1}(S),\mathcal{H}^{1}(S^{\prime})$. Up to rotation and the
action of the Teichmüller flow, we can obtain a new surface $S^{\prime}_{1}$
with no vertical saddle connection and built by zippered rectangles using a
segment of length one. Let $(\mathcal{A},\pi,\lambda_{0},\tau_{0})$ be the
data for the zippered rectangles construction. We denote
$\lambda_{0}=(\lambda_{0}(\alpha))_{\alpha}$ the vector whose entries are the
length of the intervals, and $\tau_{0}=(\tau_{0}(\alpha))_{\alpha}$ the height
of the rectangles in the zippered rectangles construction. These data allow us
to construct a pointed translation surface that is canonically isomorphic to
$S_{1}^{\prime}$. The construction also gives canonical basis
$(h_{\alpha})_{\alpha\in\mathcal{A}}$ and
$(\zeta_{\alpha})_{\alpha\in\mathcal{A}}$ of the homology groups:

$H_{1}(S\backslash\Sigma_{S^{\prime}},\mathbb{Z}),~{}~{}~{}\text{and}~{}~{}~{}H_{1}(S,\Sigma_{S^{\prime}},\mathbb{Z}),$

see [Yoc07] for more details. For all $x\in I_{\alpha}^{t}$, we have the first
return cocycle

$C_{1}(S^{\prime},x)=h_{\alpha}.$

The zippered rectangles construction provides an open map from a neighborhood
of $(\lambda_{0},\tau_{0})$ in
$\mathbb{R}_{>0}^{\mathcal{A}}\times\mathbb{R}_{>0}^{\mathcal{A}}$ to
$\widetilde{\mathcal{H}}(\kappa)$. We can choose a neighborhood $U$ of
$(\lambda_{0},\tau_{0})$ that satisfies the following properties:

  * •

There exists $\delta>0$ such that
$U=\\{g_{s}(\lambda,\tau)~{},~{}(s,(\lambda,\tau))\in(-\delta,\delta)\times
V\\}$ where $V$ is the cross section formed by surfaces of $U$ such that
$\|\lambda\|_{1}=1$ (with $g_{s}(\lambda,\tau)=(e^{s}\lambda,e^{-s}\tau)$).

  * •

There exists $C_{1}>0$ such that, for all $(\lambda,\tau)\in U$:

$ C^{-1}_{1}\leq\lambda_{\alpha}\leq
C_{1},~{}~{}~{}\text{and}~{}~{}~{}C^{-1}_{1}\leq\tau_{\alpha}\leq
C_{1},~{}~{}~{}\forall\alpha\in\mathcal{A}.$

The space $\widetilde{\mathcal{H}}(\kappa)$ of pointed translation surfaces is
a manifold, then we can reduce $V,\delta$ and assume that the map
$U\rightarrow\widetilde{\mathcal{H}}(\kappa)$ is an embedding, and then the
return time on $V$ of the Teichmüller flow is bounded from below by $2\delta$.
We still denote $U,V$ the image of $U,V$ in $\widetilde{\mathcal{H}}(\kappa)$
and $U^{(1)},V^{(1)}\subset\widetilde{\mathcal{H}}^{1}(\kappa)$. For all
surfaces $Y^{\prime}=(Y,y_{0})\in U$, there is a canonical representation of
$Y^{\prime}$ using the same model as $S^{\prime}_{1}$, and then we have
trivialization of the homology group of all surfaces in $U$. We use the
following notations:

  * •

$I(Y^{\prime})$ canonical segment of $Y^{\prime}$ used for the zippered
rectangles construction,

  * •

$w_{Y}$ is the Abelian differential defining $Y$,

  * •

$C_{\textsc{n}}(Y^{\prime},y)$ the $\textsc{n}-th$ return cycle on
$I(Y^{\prime})$ for $y\in I(Y^{\prime})$,

  * •

$C_{\textsc{n}}(Y^{\prime})$ the $\textsc{n}-th$ return cycle of the vertical
leaf starting from the marked point $y_{0}$,

  * •

$\|c\|_{1}=\sum_{\alpha}|\langle\zeta_{\alpha},c\rangle|,~{}~{}\forall c\in
H_{1}(Y\backslash\Sigma_{Y^{\prime}},\mathbb{R})$.

######  Remark 4.

For all $\textsc{n},\alpha,y$, we have
$\langle\zeta_{\alpha},C_{\textsc{n}}(Y^{\prime},y)\rangle\geq 0$, when it’s
defined.

We also need the following technical lemma, let $I^{*}(Y)$ be the subset of
points in $I(Y)$ with an infinite future orbit.

######  Lemma 3.

If the vertical flow of $S^{\prime}_{1}$ is minimal, then there exists $l$
such that all the trajectories of length larger than $l$ cross all the cycles
$\zeta_{\alpha}$ at least once, in other words.

(3.2)
$\langle\zeta_{\alpha},C_{\textsc{n}}(S^{\prime}_{1},x)\rangle>0,~{}~{}~{}~{}~{}\forall\textsc{n}\geq
l,~{}~{}~{}\forall x\in
I^{*}(S^{\prime}_{1}),~{}~{}~{}\text{and}~{}~{}~{}\forall\alpha\in\mathcal{A}.$

For all $l\geq 0$, by reducing $V$, we can assume for all $Y^{\prime}\in V$,
and for all $\textsc{n}\leq l$:

(3.3) $C_{\textsc{n}}(Y^{\prime})=C_{\textsc{n}}(S^{\prime}_{1}).$

The proof of this lemma uses the formalism of intervals exchange, and it’s
given in the appendix. By assumption, we can choose an integer $l$ that
satisfies the first part of the lemma and reduce $V$ so that the second
statement is also satisfied (we keep the same notation $V$ for this smaller
cross section). As we have a uniform lower bound for the return time in
$V^{(1)}$, the measure on $V^{(1)}$ induced by $\mu_{S^{\prime}}$ is finite.
We denote by $\sigma$ the volume of $V^{(1)}$, and $\nu$ the measure induced
on $V^{(1)}$.

The following result is a corollary of the Birkhoff theorem (theorem Theorem).

######  Corollary 1.

For almost every $\theta$, the following limit is true:

$ \lim_{s\rightarrow+\infty}\frac{\\#\\{u\leq
s,~{}~{}~{}~{}g_{u}r_{\theta}S^{\prime}\in V\\}}{s}=\sigma.$

In the LHS we count the number of visits in the cross section on the interval
$[0,s]$.

### Large excursion

For almost every $\theta$, the surface $r_{\theta}S^{\prime}$ is generic for
the corollary 1, and then the trajectory $(g_{s}r_{\theta}S^{\prime})$ visits
$V$ an infinite number of times. We denote by $(S^{(n)}_{\theta})_{n\geq 0}$
the sequence of visits and $S_{\theta}=S^{(0)}_{\theta}$ the first one, and we
still denote $f$ the cocycle on $S_{\theta}$ (we drop the ′ for simplicity).
We prove the following result, which allows us to construct trajectories with
a good diffusion rate.

######  Proposition 1.

If $\Lambda(f)>0$, for almost every $\theta$, for all $\epsilon>0$, there
exists a sequence $\textsc{n}_{n}$

(3.4) $|\langle f,C_{\textsc{n}_{n}}(S_{\theta})\rangle|\geq\exp((\Lambda
f-\epsilon)\log\textsc{n}_{n}).$

Moreover

(3.5) $\lim_{n}\frac{\log\textsc{n}_{n}}{n}=\beta:=\frac{1}{\sigma}.$

Using lemma 1 on lattice covering, we can deduce the following corollary:

######  Corollary 2.

If $\pi:\tilde{S^{\prime}}\rightarrow S^{\prime}$ is a lattice covering of a
compact translation surface defined by a cocycle $f$, with $\Lambda f>0$. For
almost every $\theta$ and for all $\epsilon>0$, there exists a sequence of
times $(t_{n})_{n\geq 1}$ such that

$\lim_{n}\frac{\log t_{n}}{n}=\frac{1}{\sigma},$

and

$\liminf_{n}\frac{\log
d(\tilde{x},\tilde{\phi}_{t_{n}}(\tilde{x},\theta))}{\log t_{n}}=\Lambda
f-\epsilon.$

Where $\tilde{x}$ is the marked point.

We give the proof of proposition 1.

###### Proof.

We fix $\theta$ In the marked Teichmüller space, we have
$S^{(n)}_{\theta}=g_{s_{n}}\Gamma^{(n)}(S_{\theta})S_{\theta}$, where
$\Gamma^{(n)}(S_{\theta})$ is a homeomorphism and $s_{n}:=s_{n}(S_{\theta})$
is the n-th return time on $V^{(1)}$ of $(g_{s}(S_{\theta}))_{s\geq 0}$. If
$I^{(n)}(S_{\theta})=e^{-s_{n}}I(S_{\theta})$, and let
$N^{(n)}_{\alpha}:=N^{(n)}_{\alpha}(S_{\theta})$ be the return time on
$I^{(n)}(S_{\theta})$ of the trajectory $(T^{k}_{S_{\theta}}y)_{k\geq 0}$ of a
point $y\in
I^{(n)}_{\alpha}(S_{\theta})=e^{-s_{n}}I_{\alpha}(S^{(n)}_{\theta})$, and
$r_{k}^{(n)}:=r_{k}^{(n)}(S_{\theta})$ the k-th return of the trajectory
starting from the marked point. We have the relation

$
\langle\zeta_{\alpha},C_{\textsc{n}}(S_{\theta},y)\rangle=|\\{k<\textsc{n},~{}T_{S_{\theta}}^{k}(y)\in
I_{\alpha}^{t}(S_{\theta})\\}|.$

And then we have

$ \|C_{\textsc{n}}(S_{\theta},y)\|_{1}=\textsc{n}.$

The pullback by $\Gamma^{(n)}(S_{\theta})$ of the first return cycle on
$I(S^{(n)}_{\theta})$ starting to a point in $y\in
I_{\alpha}(S^{(n)}_{\theta})$ is equal to the first return cycle on
$I^{n}(S_{\theta})$ starting from the point $e^{-s_{n}}y\in
I^{(n)}_{\alpha}(S_{\theta})$. In other words, we have the following equality:

$
C_{N^{(n)}_{\alpha}}(S_{\theta},e^{-s_{n}}y)=\Gamma^{(n)}(S_{\theta})^{*}C_{1}(S^{(n)}_{\theta},y)=\Gamma^{(n)}(S_{\theta})^{*}h_{\alpha},$

and also

$
C_{r_{k}^{(n)}}(S_{\theta})=\Gamma^{(n)}(S_{\theta})^{*}C_{k}(S^{(n)}_{\theta}).$

We obtain the following:

$\displaystyle N^{(n)}_{\alpha}$ $\displaystyle=$
$\displaystyle\|\Gamma^{(n)}(S_{\theta})^{*}h_{\alpha}\|_{1},~{}~{}~{} \forall
n,\alpha;$ $\displaystyle r^{(n)}_{k}$ $\displaystyle=$
$\displaystyle\|\Gamma^{(n)}(S_{\theta})^{*}C_{k}(S_{\theta}^{(n)})\|_{1},~{}~{}~{}
\forall n,k.$

For all $n$, using lemma 3 if $k\leq l$, we have

$|\langle f,C_{r^{(n)}_{k}}(S_{\theta})\rangle|=|\langle
f,\Gamma^{(n)}(S_{\theta})^{*}C_{k}(S_{\theta}^{(n)})\rangle|=|\langle\Gamma^{(n)}(S_{\theta})_{*}f,
C_{k}(S_{\theta})\rangle|.$

Using the same lemma 3 for each $\alpha$ there exists $k<l$ such as
$C_{k+1}(S_{\theta})=h_{\alpha}+C_{k}(S_{\theta})$. According to this, the
cycles $(C_{k}(S_{\theta}))_{k\leq l}$ generate the space
$H_{1}(S\backslash\Sigma_{S^{\prime}},\mathbb{R})$, then there exists a
constant $C_{2}>0$ (that depends only of the cross section in the moduli
space) such that for all cycle $c\in H_{1}(S,\Sigma_{S^{\prime}},\mathbb{R})$
we have

$\max_{k\leq l}|\langle c,C_{k}(S_{\theta})\rangle|\geq C_{2}\|c\|_{1}.$

And then, for all $n$, there exists $l_{n}:=l_{n}(S_{\theta})$ such as

$|\langle f,C_{r^{(n)}_{l_{n}}}(S_{\theta})\rangle|\geq
C_{2}\|\Gamma^{(n)}.(S_{\theta})_{*}f \|_{1}.$

As the cross section is relatively compact in the Teichmüller space, we can
compare the Hodge norm and the $\|.\|_{1}$ norm on the Hodge bundle restricted
to the cross section (by pullback, $f$ induces an element of
$H_{1}(S\backslash\Sigma_{S^{\prime}},\mathbb{R})$, and then we can define
$\|f\|_{1}=\sum_{\alpha}|\langle f,h_{\alpha}\rangle|$ on
$H^{1}(S,\mathbb{R})$). There exists a constant $C_{3}>0$ such that for all
$f\in H_{1}(S,\mathbb{R})$:

$\|f\|_{1}\geq C_{3}\|f\|_{h}.$

Let $\textsc{n}_{n}=r^{(n)}_{l_{n}}$, for almost every $\theta$, by the
theorem Theorem we have

$\frac{\log \|\Gamma^{(n)}(S_{\theta})_{*}f \|_{1}}{n}\geq\frac{\log
\|G_{s_{n}} (r_{\theta}\cdot S^{\prime},f)\|_{h}}{n}+\frac{\log C_{3}}{n}.$

Then, by applying the Oseldet theorem to $(S^{\prime},f)$, we have for almost
every $\theta$:

$\lim_{n}\frac{\log \|G_{s_{n}} r_{\theta}\cdot
S^{\prime},f)\|_{h}}{s_{n}}=\Lambda f.$

And then, as

$\lim_{n\rightarrow+\infty}\frac{s_{n}}{n}=\beta,$

we have

$\lim_{n\rightarrow+\infty}\frac{\log \|\Gamma^{(n)}(S_{\theta})_{*}f
\|_{1}}{n}=\Lambda f\beta.$

To conclude, we use a lemma extracted from [Zor99]:

######  Lemma 4.

For almost every $\theta$ and for all $\alpha$

$\lim_{n\rightarrow+\infty}\frac{\log N^{(n)}_{\alpha}(S_{\theta})}{n}=\beta.$

We can see that there is a sequence of strictly positive integer
$(a(\alpha))_{\alpha}$ independent of $n$, with

$\textsc{n}_{n}=\sum_{\alpha}a(\alpha)N_{\alpha}^{(n)}(S_{\theta})~{}~{}~{}~{}~{}~{}\sum_{\alpha}a(\alpha)=l,$

and we can use this to obtain

$l\max_{\alpha}N_{\alpha}^{(n)}\leq\textsc{n}_{n}\leq
l\min_{\alpha}N_{\alpha}^{(n)}.$

And finally, the lemma gives

$\lim_{n\rightarrow+\infty}\frac{\log\textsc{n}_{n}}{n}=\beta,$

and then we have

$\lim_{n\rightarrow+\infty}\frac{\log \|\Gamma^{(n)}(S_{\theta})_{*}f
\|_{1}}{\log\textsc{n}_{n}}=\Lambda f.$

∎

### Average of the diffusion

Here we prove theorem 2. We use the results of A. Zorich [Zor99] in order to
establish a uniform upper bound. In a second time, we use the proposition 1
and the uniform upper bound to give a lower bound, and this is enough to prove
the proposition 2.

Uniform upper bound: For the last section, we need a uniform upper bound,
which can be obtained using the same techniques as in the Zorich paper’s
[Zor99]. We do not give proof of this result here.

######  Proposition 2.

For almost all $\theta$, for all cocycle $f$ such that $\Lambda f>0$, and for
all $\epsilon>0$, there exists $N_{0}$ such that

$ \frac{\log|\langle f,C_{\textsc{n}}(S_{\theta},x)\rangle|}{\log
n}\leq\Lambda(f)+\epsilon,~{}~{}~{}\forall\textsc{n}\geq N_{0}.$

The bound is true for all $x\in I^{*}(S_{\theta})$ (i.e. points with an
infinite future orbit).

The important point in this proposition is the fact that the bound is uniform
on the segment $I^{*}(S_{\theta})$, we will use it in the next section. We can
obtain the following corollary by performing a summation of the last
inequality.

######  Corollary 3.

Let $\theta$ and $f$ that satisfy the statement of the last proposition. Then,
for all $\epsilon>0$, there exists $N_{0}$ such that

$\sum_{k\leq\textsc{n}}|\langle f,C_{k}(S_{\theta},x)\rangle|\leq
\exp\left(\left(\Lambda(f)+1+\epsilon\right)\log\textsc{n}\right),~{}~{}~{}\forall\textsc{n}\geq
N_{0},$

and the bound is true for all $x\in I^{*}(S_{\theta})$.

###### Proof.

Let $\theta$ satisfying the assumption of the last proposition. For all
$\epsilon>0$, according to the last proposition, there is $N_{0}\geq 0$ such
as: for all $\textsc{n}\geq N_{0}$ and all $x\in I^{*}(S_{\theta})$ we have

$|\langle f,C_{\textsc{n}}(S_{\theta},x)\rangle|\leq\textsc{n}^{\Lambda
f+\frac{\epsilon}{2}}$

Moreover, by using the fact that $C_{\textsc{n}}$ is an additive cocycle we
have the trivial bound:

$|\langle
f,C_{\textsc{n}}(S_{\theta},x)\rangle|\leq\textsc{n}\max_{\alpha}|\langle
f,h_{\alpha}\rangle|.$

We can write, for all $\textsc{n}\geq N_{0}$

$\displaystyle\sum_{k\leq\textsc{n}}|\langle f,C_{k}(S_{\theta},x)\rangle|$
$\displaystyle\leq$ $\displaystyle\sum_{k<N_{0}}|\langle
f,C_{k}(S_{\theta},x)\rangle|+\sum_{N_{0}\leq k\leq\textsc{n}}|\langle
f,C_{k}(S_{\theta},x)\rangle|$ $\displaystyle\leq$ $\displaystyle
N^{2}_{0}\max_{\alpha}|\langle
f,h_{\alpha}\rangle|+\sum_{k\leq\textsc{n}}k^{\Lambda f+\frac{\epsilon}{2}}$

Using comparison series vs integral, we have

$\sum_{k\leq\textsc{n}}k^{\Lambda
f+\frac{\epsilon}{2}}\leq\int_{0}^{\textsc{n}+1}x^{\Lambda
f+\frac{\epsilon}{2}}dx\leq\frac{(\textsc{n}+1)^{\Lambda
f+\frac{\epsilon}{2}+1}}{\Lambda f+\frac{\epsilon}{2}+1}.$

Then we can find two constants $A,B$ (depending of $\epsilon$ only ) such as,
for all $\textsc{n}\geq N_{0}$, we have

$\sum_{k\leq\textsc{n}}|\langle f,C_{k}(S_{\theta},x)\rangle|\leq
A+B~{}\textsc{n}^{\Lambda f+1+\frac{\epsilon}{2}}\leq(A~{}\textsc{n}^{-\Lambda
f-1-\frac{\epsilon}{2}}+B)~{}\textsc{n}^{\Lambda f+1+\frac{\epsilon}{2}}.$

If n goes to $\infty$, the factor $A~{}\textsc{n}^{-\Lambda
f-1-\frac{\epsilon}{2}}+B$ is bounded, then its smaller than
$\textsc{n}^{\frac{\epsilon}{2}}$ for n large enought. Then, by increasing
$N_{0}$, we show that: for all $\epsilon$, there is $N_{0}$ such that, for all
$\textsc{n}\geq N_{0}$ we have

$\sum_{k\leq\textsc{n}}|\langle
f,C_{k}(S_{\theta},x)\rangle|\geq\textsc{n}^{\Lambda f+1+\epsilon}.$

As before, the bound is uniform on $I^{*}(S_{\theta})$. ∎

#### Lower bound:

Now, we need to establish a lower bound for the second statement of
proposition 2.

###### Proof.

First, we exhibit intervals of discrete times on which the sum of $|\langle
f,C_{k}(S_{\theta})\rangle|$ is large enough. To do this, we use large
excursions (see proposition 1) and the fact that the diffusion is bounded
around such excursions (see corollary 3). The trajectory goes far and stays
far enough. In a second, we prove that these intervals can be used to minorate
$\frac{\log\sum_{k\leq\textsc{n}}C_{k}(S_{\theta})}{\log\textsc{n}}$ where n
belongs to some intervals of discrete times. Finally, we show that the union
of these intervals contains $N_{1}+\mathbb{N}$ for $N_{1}$ large enought.

$1)$ For all $p\geq 0$, we can write

$\sum_{k\leq\textsc{n}_{n}+p}|\langle
f,C_{k}(S_{\theta})\rangle|\geq\sum_{0\leq k\leq p}|\langle
f,C_{\textsc{n}_{n}+k}(S_{\theta})\rangle|. $

Using the fact that $C_{\textsc{n}}(S_{\theta},x)$ is an additive cocycle for
the interval exchange $T_{S_{\theta}}$, we have:

$|\langle f,C_{\textsc{n}_{n}+k}(S_{\theta})\rangle|\geq|\langle
f,C_{\textsc{n}_{n}}(S_{\theta})\rangle|-|\langle
f,C_{k}(S_{\theta},T_{S_{\theta}}^{\textsc{n}_{n}}x)\rangle|. $

Of course, if $x$ has an infinite future orbit, then it is also true for
$T_{S_{\theta}}^{\textsc{n}_{n}}x$. We can apply proposition 3 for an
$\epsilon$ such that $0<2\epsilon<\text{min}\\{\beta,\Lambda f\\}$, we can
find $p_{\epsilon}$ which does not depend on $x,n$ and such that, for all
$p\geq p_{\epsilon}$

$\sum_{0\leq k\leq p} |\langle
f,C_{k}(S_{\theta},T_{S_{\theta}}^{\textsc{n}_{n}}x)\rangle|\leq\exp\left((\Lambda
f+1+\epsilon)\log p\right). $

Using the lower bound of proposition 1 and the lower bound on
$\textsc{n}_{n}$, there is $n_{\epsilon}$ such that, for all $n\geq
n_{\epsilon}$:

$\displaystyle \sum_{k\leq\textsc{n}_{n}+p}|\langle
f,C_{k}(S_{\theta},x)\rangle|$ $\displaystyle\geq$ $\displaystyle
p\exp\left((\Lambda f-\epsilon)\textsc{n}_{n}\right)-\exp\left((\Lambda
f+1+\epsilon)\log p\right)$ $\displaystyle $ $\displaystyle\geq$
$\displaystyle p\exp\left((\Lambda
f-\epsilon)(\beta-\epsilon)n\right)-\exp\left((\Lambda f+1+\epsilon)\log
p\right)$ $\displaystyle\geq$ $\displaystyle p\left(\exp\left((\Lambda
f-\epsilon)(\beta-\epsilon)n\right)-\exp\left((\Lambda f+\epsilon)\log
p\right)\right).$

If $p\leq 2^{-\frac{1}{\Lambda f+\epsilon}}\exp\left(\frac{\Lambda
f-\epsilon}{\Lambda f+\epsilon}(\beta-\epsilon)n\right)$, we then have

$\exp\left((\Lambda f-\epsilon)(\beta-\epsilon)n\right)-\exp\left((\Lambda
f+\epsilon)\log p\right)\geq\frac{\exp\left((\Lambda
f-\epsilon)(\beta-\epsilon)n\right)}{2}$

Let $p_{n}$ be the integer part of $\exp(-\frac{\log(2)}{\Lambda
f+\epsilon})\exp\left(\frac{\Lambda f-\epsilon}{\Lambda
f+\epsilon}(\beta-\epsilon)n\right)$. As $p_{n}$ goes to $\infty$ when $n$
grows up, there is $n_{\epsilon}^{\prime}\geq n_{\epsilon}$ and $A_{2}>0$ such
that if $n\geq n^{\prime}_{\epsilon}$, we have $p_{n}\geq n_{\epsilon}$. Then,
we have:

$\displaystyle \sum_{k\leq\textsc{n}_{n}+p_{n}}|\langle
f,C_{k}(S_{\theta})\rangle| $ $\displaystyle\geq$ $\displaystyle
\frac{p_{n}}{2}\exp\left((\Lambda f-\epsilon)(\beta-\epsilon)n\right)$
$\displaystyle $ $\displaystyle\geq$ $\displaystyle \left(2^{-\frac{1}{\Lambda
f+\epsilon}}\exp\left(\frac{\Lambda f-\epsilon}{\Lambda
f+\epsilon}(\beta-\epsilon)n\right)-1\right) \exp\left( (\Lambda
f-\epsilon)(\beta-\epsilon)n\right)$ $\displaystyle\geq$
$\displaystyle\left(2^{-\frac{1}{\Lambda f+\epsilon}}-\exp\left(-\frac{\Lambda
f-\epsilon}{\Lambda
f+\epsilon}(\beta-\epsilon)n\right)\right)\exp\left(\left(\Lambda
f+\frac{\Lambda f-\epsilon}{\Lambda
f+\epsilon}-\epsilon\right)(\beta-\epsilon)n\right).$ $\displaystyle\geq$
$\displaystyle A_{2}\exp\left(\left(\Lambda f+\frac{\Lambda
f-\epsilon}{\Lambda f+\epsilon}-\epsilon\right)(\beta-\epsilon)n\right).$

Finally obtain

$\displaystyle \log\sum_{k\leq\textsc{n}_{n}+p_{n}}|\langle
f,C_{k}(S_{\theta})\rangle|)$ $\displaystyle\geq$
$\displaystyle\log\sum_{\textsc{n}_{n}<k\leq\textsc{n}_{n}+p_{n}}|\langle
f,C_{k}(S_{\theta})\rangle|)$ $\displaystyle\geq$ $\displaystyle\left(\Lambda
f+\frac{\Lambda f-\epsilon}{\Lambda
f+\epsilon}-\epsilon\right)(\beta-\epsilon)n +\log A_{2}.$

$2)$ Using the expression of $p_{n}$, there exist constants $A_{3}$
independent of $\epsilon,n$ such that

$p_{n}\leq A_{3}\exp\left(\frac{\Lambda f-\epsilon}{\Lambda
f+\epsilon}(\beta-\epsilon)n\right)\leq A_{3}\exp((\beta+\epsilon)n).$

If $n$ is large enough, we have obviously

$
\frac{\log(A_{3}+1)}{n+1}\leq\epsilon,~{}~{}~{}\text{and}~{}~{}~{}\frac{\beta-\epsilon}{n+1}\leq\epsilon.$

And then

$\displaystyle \frac{(\beta-\epsilon)n}{\log(\textsc{n}_{n+1}+p_{n+1})}$
$\displaystyle\geq$
$\displaystyle\frac{(\beta-\epsilon)n}{\log(A_{3}\exp((\beta-\epsilon)(n+1))+\textsc{n}_{n+1})}$
$\displaystyle\geq$
$\displaystyle\frac{(\beta-\epsilon)n}{\log(A_{3}+1)+(\beta+\epsilon)(n+1)}$
$\displaystyle\geq$ $\displaystyle\frac{\beta-2\epsilon}{2\epsilon+ \beta}.$

Then, for all n such as $\textsc{n}_{n}+p_{n}\leq\textsc{n}$ and
$\textsc{n}\leq\textsc{n}_{n+1}+p_{n+1}$, we have:

$\displaystyle \frac{\log\sum_{k\leq\textsc{n}}|\langle
f,C_{k}(S_{\theta})\rangle|}{\log\textsc{n}} $ $\displaystyle\geq$
$\displaystyle \frac{\log\sum_{k\leq\textsc{n}_{n}+p_{n}}|\langle
f,C_{k}(S_{\theta})\rangle|}{\log(\textsc{n}_{n+1}+p_{n+1})}$ $\displaystyle $
$\displaystyle\geq$
$\displaystyle\frac{n(\beta-\epsilon)}{\log(\textsc{n}_{n+1}+p_{n+1})}\left(\Lambda
f+\frac{\Lambda f-\epsilon}{\Lambda f+\epsilon}-\epsilon\right) +\frac{\log
A_{2}}{\log(\textsc{n}_{n+1}+p_{n+1})}$ $\displaystyle $ $\displaystyle\geq$
$\displaystyle \frac{\beta-2\epsilon}{2\epsilon+ \beta}
\left(\Lambda(f)+\frac{\Lambda f-\epsilon}{\Lambda f+\epsilon}-\epsilon\right)
+\frac{\log A_{2}}{\log(\textsc{n}_{n+1}+p_{n+1})}.$

As it goes to zeros when $n$ is large, the last term is useless; we can find
$n_{\epsilon}^{\prime\prime}\geq n_{\epsilon}^{\prime}$ such that, for all
$n\geq n_{\epsilon}^{\prime\prime}$ and for all n that satisfies

$
\textsc{n}_{n}+p_{n}\leq\textsc{n}~{}\text{and}~{}\textsc{n}\leq\textsc{n}_{n+1}+p_{n+1},$

we have

$ \frac{\log\sum_{k\leq\textsc{n}}|\langle
f,C_{k}(S_{\theta})\rangle|}{\log\textsc{n}}\geq
\frac{\beta-2\epsilon}{2\epsilon+\beta} \left(\Lambda f+\frac{\Lambda
f-\epsilon}{\Lambda f+\epsilon}-2\epsilon\right).$

$3)$ It remains to prove that the bound is true for all n big enough. Let
$\textsc{n}\geq
N_{\epsilon}:=\textsc{n}_{n_{\epsilon}^{\prime\prime}}+p_{n_{\epsilon}^{\prime\prime}}$,
and $n_{0}$ be the smaller integer bigger than $n_{\epsilon}^{\prime\prime}+1$
such that $\textsc{n}\leq\textsc{n}_{n_{0}}+p_{n_{0}}$. We must have
$\textsc{n}\geq \textsc{n}_{n_{0}-1}+p_{n_{0}-1}$, and then the last
inequality is true for this n. Then it’s true for all $\textsc{n}\geq
N_{\epsilon}$, and we can conclude that

$ \liminf_{\textsc{n}} \frac{\log\sum_{k\leq\textsc{n}}|\langle
f,C_{k}(S_{\theta})\rangle|}{\log\textsc{n}} \geq \frac{\beta
-2\epsilon}{2\epsilon+ \beta} \left(\Lambda f+\frac{\Lambda
f-2\epsilon}{\Lambda f+\epsilon}-2\epsilon\right).$

To conclude, the bound is valid for all $\epsilon>0$ small enough. Finally,
when $\epsilon$ goes to zero, we obtain the claim:

$ \liminf_{\textsc{n}} \frac{\log\sum_{k\leq\textsc{n}}|\langle
f,C_{k}(S_{\theta})\rangle|}{\log\textsc{n}} \geq \Lambda f+1.$

Because the LHS tends to $\Lambda f+1$ when $\epsilon$ goes to $0$. ∎

#### End of the proof of theorem 2

We achieve the proof of the theorem 2.

###### Proof.

Let $\theta$ such as $S^{\prime}_{\frac{\pi}{2}-\theta}$ is generic for both
theorems Theorem and Theorem. By using lemma 3, the lower bound and remark 2
we obtain

$\lim_{\textsc{n}\rightarrow\infty}\frac{\log\sum_{k\leq\textsc{n}}\|\langle
f,C_{k}(S^{\prime}_{\theta})\|}{\log(\textsc{n})}=\Lambda f+1$

Then, by using lemma 2 we can conclude that

$\lim_{T\rightarrow\infty}\frac{\log\int_{0}^{T}d_{\tilde{S}}(\tilde{x},\tilde{\phi}_{t}^{\theta}(\tilde{x})}{\log(T)}=\Lambda
f+1.$

Where $\tilde{\phi}_{t}(\tilde{x})$ is the flow on $\tilde{S}$ in the
direction $\theta$. Then, for all $\tilde{x}$ and almost all $\theta$, the
statement of theorem 2 is true. ∎

######  Remark 5.

To be correct, we do not really prove the theorem for
$r_{\theta}\cdot\tilde{S}^{\prime}$ but for $\tilde{S}^{\prime}_{\theta}$. But
the two differ by the action of the Teichmüller flow, and this does not affect
the statement of the theorem.

Using this theorem and the previous works on billiard [DHL14], we can deduce a
similar statement for the wind tree model. This step is now classical in the
theory of billiards.

##  Appendix A Proof of lemma 3

An intervals exchange of $n$ intervals can be defined by data
$(\mathcal{A},\pi^{t},\pi^{b},\lambda)$, where
$\pi^{\cdot}:\mathcal{A}\to\llbracket 1,n\rrbracket$. Assume that the data are
irreducible (see [Via08]), and we denote $(T_{\lambda},I_{\lambda})$ the
corresponding intervals exchange. As before, let $I^{*}_{\lambda}$ be the
points in $I_{\lambda}$ that have an infinite future orbit, i.e., the points
that never hit a singularity. We denote by $\iota_{\lambda}(x)$ the sequence
of $(\alpha_{n})$ such that $T^{n}_{\lambda}(x)\in I^{t}_{\alpha_{n}}$, it is
well defined on $I^{*}_{\lambda}$. It is easy to see that this function is
continuous for the product topology,
$\iota_{\lambda}\in\mathcal{A}^{\mathbb{N}}$. We denote $\Sigma_{\lambda}$ the
closure of $\iota_{\lambda}(I^{*}_{\lambda})$ in $\mathcal{A}^{\mathbb{N}}$.

######  Lemma 5.

The interval exchange is conjugated to the shift on $\Sigma_{\lambda}$. We
have the following diagram, where $h$ is an almost everywhere homeomorphism:

$\textstyle{\Sigma_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{h}$$\textstyle{\Sigma_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{
I_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{I_{\lambda}}$

By using this lemma, we can prove the first statement of lemma 3.

###### Proof.

If $T_{\lambda}$ is a minimal interval exchange, then for all $x\in
I^{*}_{\lambda}$, the future orbit of $x$ is dense in $I_{\lambda}$. And then
the function

$ m_{\alpha}(x)=\min\\{n,T^{n}_{\lambda}x\in I_{\lambda,\alpha}\\},$

is well defined on $I^{*}_{\lambda}$. It is easy to check that the function

$ \tilde{m}_{\alpha}(u)=\min\\{n,\alpha_{n}=\alpha\\}$

is a continuous function on $\Sigma_{\lambda}$ and also
$m_{\alpha}(x)=\tilde{m}_{\alpha}(\iota_{\lambda}(x))$. So, as
$\Sigma_{\lambda}$ is compact, then $\tilde{m}_{\alpha}$ is bounded on
$\Sigma_{\lambda}$, and then $\max_{\alpha}m_{\alpha}$ is bounded on
$I^{*}_{\lambda}$. ∎

Now we prove the second part; more precisely, we will prove the following
lemma:

######  Lemma 6.

For all intervals exchange $T_{\lambda_{0}}$ with no connection, for all $m$,
there exists a neighborhood $V$ of $\lambda_{0}$ such that

$ \iota_{\lambda}(0)_{n}=\iota_{\lambda_{0}}(0)_{m}~{}~{}~{}~{}~{}~{}\forall
n<m~{},~{}\forall\lambda\in V$

###### Proof.

We will prove this result by induction on $n$, assume that’s true for $m$, and
let $V$ be the neighborhood of $\lambda$. We need to introduce some notations,
i.e., let $\alpha_{n}=\iota_{\lambda_{0}}(0)_{n}$ and $\gamma=\alpha_{m+1}$.
By assumption, we can write:

$ T_{\lambda}^{m+1}(y)=\sum_{0\leq k\leq
m}\delta_{\iota_{\lambda}(y)}(\lambda)=\sum_{0\leq k\leq
m}\delta_{\alpha_{k}}(\lambda)~{}~{}~{}~{}\forall y\in]0,u[.$

If $\phi(\lambda)=T_{\lambda}^{m+1}(0)=\sum_{0\leq k\leq
N}\delta_{\alpha_{k}}(\lambda)$, where $\delta_{\alpha_{k}}(\lambda)$ is the
translation from $I^{t}_{\lambda,\alpha}$ to $I^{b}_{\lambda,\alpha}$. It is
defined by the following formula:

$
\delta_{\alpha}(\lambda)=\underset{\pi^{t}(\beta)\geq\pi^{t}(\alpha)}{\sum}\lambda_{\beta}-\underset{\pi^{b}(\beta)\geq\pi^{b}(\alpha)}{\sum}\lambda_{\beta}.$

And then $\phi$ is continuous in $\lambda$. If we denote
$\epsilon=\frac{d(\phi(\lambda_{0}),(I_{\lambda_{0},\gamma}^{t})^{c})}{3}$, it
exists $V^{\prime}$ a neighborhood of $\lambda_{0}$ in $V$ such that

$ |\phi(\lambda)-\phi(\lambda_{0})|\leq\epsilon~{}~{}~{}~{}\forall\lambda\in
V^{\prime}.$

Let $u^{+}_{\alpha}(\lambda),u^{-}_{\alpha}(\lambda)$ such that
$I^{t}_{\lambda,\alpha}=]u^{-}_{\alpha}(\lambda),u^{+}_{\alpha}(\lambda)[$, we
can write

$
u^{-}_{\alpha}(\lambda)=\underset{\pi^{t}(\beta)<\pi^{t}(\alpha)}{\sum}\lambda_{\beta},~{}~{}~{}\text{and}~{}~{}~{}u^{+}_{\alpha}(\lambda)=u^{-}_{\alpha}(\lambda)+\lambda_{\alpha}.$

The two functions are continuous, and then we can find an open neighborhood
$V^{\prime\prime}$ such that $\forall\lambda\in V^{\prime\prime}$

$u^{-}_{\gamma}(\lambda) \leq u^{-}_{\gamma}(\lambda_{0})+\epsilon
<u^{+}_{\gamma}(\lambda_{0})-\epsilon\leq u^{+}_{\gamma}(\lambda).$

And then

$T_{\lambda}^{m+1}(0)~{}\in~{}] T_{\lambda_{0}}^{m+1}(0)-\epsilon,
T_{\lambda_{0}}^{m+1}(0)+\epsilon[~{}\subset~{}]u^{-}_{\gamma}(\lambda_{0})+\epsilon
,u^{+}_{\gamma}(\lambda_{0})-\epsilon[~{}\subset~{}]u^{-}_{\gamma}(\lambda),u^{+}_{\gamma}(\lambda)[$

and then $\iota_{\lambda}(0)_{m+1}=\iota_{\lambda_{0}}(0)_{m+1}$ for all
$V^{\prime\prime}$ ∎

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