# Statistical properties of Lorentz gases on aperiodic tilings, Part 1

Rodrigo Treviño  University of Maryland, College Park<EMAIL_ADDRESS>and
Agnieszka Zelerowicz  University of Maryland, College Park<EMAIL_ADDRESS>
###### Abstract.

We consider the Lorentz gas model of category A (that is, with no corners and
of finite horizon) on aperiodic repetitive tilings of $\mathbb{R}^{2}$ of
finite local complexity. We show that the compact factor of the collision map
has the K property, from which we derive mixing for pattern-equivariant
functions as well as the planar ergodicity of the Lorentz gas flow.

##  1\. Introduction and statement of results

The Lorentz gas was originally introduced as a model for the movement of
electrons in crystals. Throughout most of the 20th century, the term crystal
referred to a solid with a periodic atomic structure. Following Shechtman’s
discovery of quasicrystals, there is now consensus [Lif07] that crystal is any
substrate whose atomic structure has long range order, which includes not only
materials whose atomic structure is periodic but also quasicrystals. These are
materials whose atomic structure is not modeled by any periodic structure but
that nonetheless exhibits long range order.

The study of Lorentz gases has a long history and rich theory and has been the
motivation for the development of many tools in ergodic theory and
mathematical physics. The list of surveys covering the history and development
is long; here we just mention some: [CM06, Szá08, Gol12, Det14]. Beyond
periodic media, statistical properties of the Lorentz gases for random
configurations of scatterers have also been obtained (see e.g. [CLS10, AL20]).
One of the biggests gaps in our knowledge of Lorentz gases is the case of
aperiodic media of low complexity, as stated in [Szá08, Mar14]. The only
rigorous work to date is the work of Marklof-Strömbergsson [MS14], which
derives limit theorems for the so-called Boltzman-grad limit for
quasicrystals, that is, studies of the low density limit in aperiodic media
with long range interactions.

This is the first paper in a program to study the statistical properties of
Lorentz gases in general aperiodic repetitive media of low complexity which
aims to close this gap. Our goal is to be broad in the sense of not assuming
much beyond aperiodicity, repetitivity and finite local complexity. In
particular we do not assume that the medium has long-range order (i.e. is not
necessarily a quasicrystal). In this paper we further assume that the Lorentz
gas belongs to category A in the classification of Chernov-Markarian [CM06],
meaning that it has finite horizon and no corners. The assumption on finite
horizon indicates that we treat cases far from the zero-density limit and
therefore our setting is different from [MS14]. The novelty of our approach is
the use of tiling spaces to obtain a compact model of the aperiodic Lorentz
gas on the plane. In the case of a periodic Lorentz gas the corresponding
compact model is given by a Sinai billiard and is a uniformly hyperbolic
diffeomorphism with singularities. Unlike Sinai billiards, the compact model
we obtain by using tiling spaces is not defined on a manifold (our space
locally is a product of a Euclidean ball and a Cantor set), hence cannot be
thought of as a diffeomorphism. On the other hand, our model exhibits
essential features common to partially hyperbolic systems. This makes the
problem qualitatively different from the periodic case.

###  1.1. Statement of results

Before stating our results, we describe the precise setting in which we work.
A tiling $\mathcal{T}$ of $\mathbb{R}^{d}$ is a collection of compact,
connected sets, called tiles, such that their union covers $\mathbb{R}^{d}$
and such that any two distinct tiles in the tiling may only intersect at their
boundaries (see §2 for background on tilings). We refer to a finite union
$\mathcal{P}$ of tiles of $\mathcal{T}$ as a patch of $\mathcal{T}$ (see
Figure 1 for an example). A tiling is repetitive if given any patch
$\mathcal{P}$ of $\mathcal{T}$ there exists an $R>0$ such that any ball of
radius $R$ in $\mathbb{R}^{d}$ contains a copy of $\mathcal{P}$ in the portion
of $\mathcal{T}$ inside this ball. It has finite local complexity if for any
$R>0$ there is a finite list of patches of diameter at most $R$ that occur in
the tiling.

Given a repetitive, aperiodic tiling $\mathcal{T}$ of finite local complexity,
there is a compact metric space $\Omega$ called the tiling space of
$\mathcal{T}$ which contains all tilings which are locally indistinguishable
from translations of $\mathcal{T}$. Assuming that $\mathcal{T}$ is repetitive
and has finite local complexity, it is one of the continuum of tilings in
$\Omega$. The space $\Omega$ is not a manifold but has the local product
structure of an open Euclidean ball and a Cantor set. There is a minimal
action of $\mathbb{R}^{d}$ on this space, making $\Omega$ a foliated space,
foliated by orbits of this action.

Figure 1. A patch of the half-hex aperiodic tiling.

If $\mathcal{T}$ is a tiling of $\mathbb{R}^{d}$, a function
$g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is $\mathcal{T}$-equivariant (or
pattern-equivariant) if there exists an $R>0$ such that if for some
$x_{1},x_{2}\in\mathbb{R}^{d}$, the largest patches $\mathcal{P}_{1}$,
$\mathcal{P}_{2}$ contained in $B_{R}(x_{1})$ and $B_{R}(x_{2})$,
respectively, are translation equivalent (that is, $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ are translated copies of each other), then
$g(x_{1})=g(x_{2})$. In other words, the value of $g(x)$ depends only on the
local pattern around $x$. More generally, if $X$ is any set and $\mathcal{T}$
is a tiling of $\mathbb{R}^{d}$, a function $g:\mathbb{R}^{d}\times
X\rightarrow\mathbb{R}$ is said to be $\mathcal{T}$-equivariant if for any
$z\in X$ the function $g_{z}(x):=g(x,z)$ is $\mathcal{T}$-equivariant.

Pattern equivariant functions are in a sense the most natural class of
functions to consider because they are closely related to functions on the
tiling space: a function on $\mathbb{R}^{d}$ is $\mathcal{T}$ equivariant if
and only if there is a transversally locally constant (see §2.2 for more
details) function $h:\Omega\rightarrow\mathbb{R}$ such that
$g(v)=h\circ\varphi_{v}(\mathcal{T})$, where $\varphi$ is the translation
action on the tiling space $\Omega$.

For a collection $\mathcal{S}\subset\mathbb{R}^{d}$ of topological balls with
$C^{3}$ boundaries, called scatterers, the Lorentz gas is the system defined
for initial conditions in
$(\mathbb{R}^{d}\setminus\overline{\mathcal{S}})\times S^{d-1}_{+}$ as the
free flight of a particle in $\mathbb{R}^{d}\setminus\overline{\mathcal{S}}$
in the direction determined by a point in $S^{d-1}_{+}$ and bouncing
elastically off $\partial\mathcal{S}$. This system has finite horizon if there
is a global constant $M>0$ such that the time between any two collisions with
$\partial\mathcal{S}$ is bounded above by $M$; it has no corners if any two
scatterers are separated; it is dispersive if every connected component of
$\overline{\mathcal{S}}$ (that is, a specific scatterer) is strictly convex.
We refer to a collection of dispersive scatterers (and to the corresponding
Lorentz gas) of finite horizon and with no corners as being of category A.

Given a tiling $\mathcal{T}$ of $\mathbb{R}^{d}$, a collection of scatters
$\mathcal{S}$ is $\mathcal{T}$-equivariant if there exists an $R>0$ such that
if $x_{1},x_{2}\in\mathbb{R}^{d}$ are the centers of mass of two scatterers
$S_{1},S_{2}\subset\mathcal{S}$ and the largest patches $\mathcal{P}_{1}$,
$\mathcal{P}_{2}$ contained in $B_{R}(x_{1})$ and $B_{R}(x_{2})$,
respectively, are translation equivalent, then the scatterers $S_{1}$ and
$S_{2}$ are translation equivalent. In other words, the geometry of a
scatterer and relative position of scatterers surrounding it in a
$\mathcal{T}$-equivariant collection of scatterers depends only on the local
pattern around it. An example of this would be the collection of scaterrers
centered at the vertex of every tile in a tiling. Another example is depicted
in Figure 2. Given a $\mathcal{T}$-equivariant collection of scatterers
$\mathcal{S}\subset\mathbb{R}^{d}$, a function
$g:\partial\mathcal{S}\rightarrow\mathbb{R}$ is $\mathcal{T}$-equivariant if
it is the restriction to $\partial\mathcal{S}$ of a $\mathcal{T}$-equivariant
function on $\mathbb{R}^{d}$.

Figure 2. An aperiodic Lorentz gas configuration. The scatterers depend only
on local configurations of the half-hex tiling in Figure 1.

Tilings as coverings of $\mathbb{R}^{d}$ have their discrete counterpart
called Delone multisets. Roughly speaking, if one thinks of a tiling as a set
of tiles, one can think of the corresponding Delone multiset as a set of
centers of masses of those tiles. We give a precise definition of a Delone
multiset and related notions in Section 2.1. Even though we state our main
results in the language of tilings, it is sometimes convenient to think of the
corresponding Delone multisets instead. In particular, given a Delone multiset
one can easily define the collection of scatterers by taking each point in a
Delone multiset as a center of mass of a scatterer (see Section 3.1.2).

Given a repetitive, aperiodic tiling $\mathcal{T}$ of finite local complexity
and a $\mathcal{T}$-equivariant collection of scatterers $\mathcal{S}$, we can
take $\partial\mathcal{S}\times S^{d-1}_{+}$ as a Poincaré section for the
Lorentz gas. This set can be given a metric topology which naturally
identifies it as a subset of $\mathcal{D}=\partial\Sigma\times
S^{d-1}_{+}\subset\Omega\times S^{d-1}_{+}$. The set $\mathcal{D}$ is then the
Poincaré section to the continuum of Lorentz gases defined in all tilings in
$\Omega$ as long as we use the same local dependence on scatterers which
defined $\mathcal{S}$. The set $\mathcal{D}$ is a compact metric space where a
global Poincaré map $f:\mathcal{D}\rightarrow\mathcal{D}$ is defined. Given
any $\mathbb{R}^{d}$-invariant measure on $\Omega$, there is a naturally
derived measure $\bar{\mu}$ on $\mathcal{D}$ which is $f$-invariant. Our first
result is for the planar Lorentz gas on aperiodic tilings (see Theorem 9 in
§5).

######  Theorem 1.

Let $\mathcal{T}$ be a repetitive, aperiodic tiling of $\mathbb{R}^{2}$ of
finite local complexity, $\mathcal{S}$ be a $\mathcal{T}$-equivariant
collection of dispersive scatterers with $C^{3}$ boundary defining a Lorentz
gas of category A, and $\mu$ an $\mathbb{R}^{2}$-invariant ergodic probability
measure on $\Omega_{\mathcal{T}}$. Then the map $f:\mathcal{D}\to\mathcal{D}$
has the K property with respect to $\bar{\mu}$.

######  Remark 1.

Theorem 1 brings an entire new class of examples of K systems to the theory of
dynamical systems. We do not know under which conditions these systems are
Bernoulli. We are currently pursuing this question.

We briefly outline the proof of this theorem. We consider measurable
partitions of $\mathcal{D}$ into local stable and unstable manifolds
respectively. The celebrated fundamental theorem of Sinai for dispersive
planar billiards holds in our case and provides a good control over the sizes
of local stable and unstable manifolds. Following arguments identical to the
proof of local ergodicity for periodic Sinai billiards, we obtain that
Lebesgue-almost every two points on the collision space of a single scatterer
can be connected by a stable-untable path. It follows that $\bar{\mu}$-almost
every connected component (a single scatterer along with all of its
incoming/outgoing collisions) is fully contained (up to a subset of Lebesgue
measure zero) in a single element of the Pinsker partition. Using the
(minimal) foliated structure of the tiling space and the dynamics of $f$, it
will follow that a dense union of connected components is contained in an
element of the Pinsker partition. Finally, using again the structure of the
tiling space and the ergodicity of $\mu$ (with respect to the translation
$\mathbb{R}^{2}$-action) we will show that the Pinsker partition is trivial.

From this mixing result on the compact system $(f,\mathcal{D},\bar{\mu})$, we
obtain a mixing result for planar Lorentz gases of category A. That is for a
$\mathcal{T}$-equivariant collection of scatterers
$\mathcal{S}\subset\mathbb{R}^{2}$, $\theta\in(-\pi/2,\pi/2)$, and
$n\in\mathbb{N}$ we denote by
$F^{n}_{\theta}:\partial\mathcal{S}\rightarrow\partial\mathcal{S}$ the map
given by the first $n$ colisions of the planar Lorentz gas with initial angle
$\theta\in(-\pi/2,\pi/2)$. The map $F^{n}_{\theta}$ is not defined on all of
$\partial\mathcal{S}\times(-\pi/2,\pi/2)$ due to discontinuities, but this is
a set of Lebesgue measure zero for all $n\in\mathbb{N}$.

Let $g:\partial\mathcal{S}\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ be a
locally integrable, $\mathcal{T}$-equivariant function. We define the planar
average of $g$ as the following limit

$\beta(g):=\lim_{T\rightarrow\infty}\frac{1}{\mbox{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(z)\cap\partial\mathcal{S}}g\,d\rho\,\cos\theta\,d\theta,$

where $\rho$ denotes the arc-length and $\theta\in(-\pi/2,\pi/2)$ is the
angle. By Lemma 7, for every locally integrable, $\mathcal{T}$-equivariant
function $g$ the above limit exists and does not depend on $z$. We denote the
set of $\mathcal{T}$-equivariant, locally integrable functions
$g:\partial\mathcal{S}\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ by
$\Delta_{\mathcal{T}}(\mathcal{S})$.

Let $\Omega$ be the tiling space of an aperiodic, repetitive tiling
$\mathcal{T}^{\prime}$ of $\mathbb{R}^{2}$ of finite local complexity, and let
$\mu$ be an invariant ergodic probability measure for the $\mathbb{R}^{2}$
action. Let $\partial\Sigma\subset\Omega$ be a transversally locally constant
set which defines for every $\mathcal{T}\in\Omega$ a $\mathcal{T}$-equivariant
collection $\mathcal{S}_{\mathcal{T}}\subset\mathbb{R}^{2}$ of dispersive
scatterers of Category A with $C^{3}$ boundary.

######  Theorem 2 (Planar averaged mixing; see Theorem 11 in §6).

For $\mu$-almost every $\mathcal{T}\in\Omega$, for any two locally integrable,
$\mathcal{T}$-equivariant functions
$g_{1},g_{2}:\partial\mathcal{S}\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$
there exist transversally locally constant functions
$h_{1},h_{2}:\mathcal{D}\rightarrow\mathbb{R}$ such that for any
$n\in\mathbb{N}$ and all $T$ large enough (depending on $n$), we have that for
any $x\in\mathbb{R}^{2}$

(1)
$\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F^{n}_{\theta}\cdot
g_{2}\,d\rho\,\cos\theta\,d\theta=\mathrm{Vol}(B_{T})\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}+o_{n}(T^{2}),$

from which we obtain the convergence

(2)
$\lim_{n\rightarrow\infty}\lim_{T\rightarrow\infty}\frac{1}{\mathrm{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F_{\theta}^{n}\cdot
g_{2}\,d\rho\,\cos\theta\,d\theta=\beta(g_{1})\beta(g_{2}).$

If the action of $\mathbb{R}^{2}$ on $\Omega$ is uniquely ergodic, the above
result can be strengthen to obtain (1) and (2) for every, and not just
$\mu$-almost every $\mathcal{T}\in\Omega$. Unique ergodicity of the
$\mathbb{R}^{2}$\- action on the tiling space $\Omega_{\mathcal{T}^{\prime}}$
generated by $\mathcal{T}^{\prime}$ is equivalent to the tiling
$\mathcal{T}^{\prime}$ having uniform patch frequency. That is, the asymptotic
frequency (14) of a given patch in $\mathcal{T}$ being the same for every
tiling $\mathcal{T}\in\Omega_{\mathcal{T}^{\prime}}$.

######  Corollary 1.

Let $\Omega$ be the tiling space of an aperiodic, repetitive tiling
$\mathcal{T}^{\prime}$ of $\mathbb{R}^{2}$ of finite local complexity and
uniform patch frequency, and let $\mu$ be the unique measure invariant under
the $\mathbb{R}^{2}$ action. Let $\partial\Sigma\subset\Omega$ be a
transversally locally constant set which defines for every
$\mathcal{T}\in\Omega$ a $\mathcal{T}$-equivariant collection
$\mathcal{S}_{\mathcal{T}}\subset\mathbb{R}^{2}$ of dispersive scatterers of
category A with $C^{3}$ boundary.

For every $\mathcal{T}\in\Omega$, for any two locally integrable,
$\mathcal{T}$-equivariant functions
$g_{1},g_{2}:\mathbb{R}^{2}\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ there
exist transversally locally constant functions
$h_{1},h_{2}:\mathcal{D}\rightarrow\mathbb{R}$ such that for any
$n\in\mathbb{N}$ and all $T$ large enough (depending on $n$), we have that for
any $x\in\mathbb{R}^{2}$

(3)
$\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F^{n}_{\theta}\cdot
g_{2}\,d\rho\,\cos\theta\,d\theta=\mathrm{Vol}(B_{T})\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}+o_{n}(T^{2}),$

from which we obtain the convergence

(4)
$\lim_{n\rightarrow\infty}\lim_{T\rightarrow\infty}\frac{1}{\mathrm{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F_{\theta}^{n}\cdot g_{2}\,dt\,\cos\theta\,d\theta=\beta(g_{1})\beta(g_{2}).$

The estimate on the error term in Theorem 2 can be sharpened in the case of
substitution tilings (also called self-similar tilings) which we introduce in
Section 2 (see Definition 1). Namely, let $\Omega$ be the tiling space of a
self-similar tiling $\mathcal{T}^{\prime}$ of $\mathbb{R}^{2}$ of finite local
complexity, and let $\mu$ be the unique measure invariant under the
$\mathbb{R}^{2}$ action. Let $\partial\Sigma\subset\Omega$ be a transversally
locally constant set which defines for every $\mathcal{T}\in\Omega$ a
$\mathcal{T}$-equivariant collection
$\mathcal{S}_{\mathcal{T}}\subset\mathbb{R}^{2}$ of dispersive scatterers of
category A with $C^{3}$ boundaries. We have the following.

######  Theorem 3 (See Theorem 12 in §6).

For any $n$ there exist $d^{+}\geq 1$ functions
$C_{i}^{n}:\Delta_{\mathcal{T}}(\mathcal{S})\times\Delta_{\mathcal{T}}(\mathcal{S})\rightarrow\mathbb{R}$
(where $d^{+}$ depends on the spectrum of the substitution matrix) such that
for any $\mathcal{T}\in\Omega$, for
$g_{1},g_{2}\in\Delta_{\mathcal{T}}(\mathcal{S}_{\mathcal{T}})$ there exist
transversally locally constant functions
$h_{1},h_{2}:\mathcal{D}\rightarrow\mathbb{R}$ such that for any
$n\in\mathbb{N}$ and all $T$ large enough (depending on $n$), we have that

(5)
$\begin{split}&\int_{-\pi/2}^{\pi/2}\int_{B_{T}\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F^{n}_{\theta}\cdot g_{2}\,d\rho\,\cos\theta\,d\theta\\\ &\hskip
36.135pt=\sum_{i=1}^{d^{+}}L_{i}(T)T^{2\frac{\log|\lambda_{i}|}{\log\lambda_{1}}}C_{i}^{n}(g_{1},g_{2})+\mathcal{O}_{n}(T),\\\
&\hskip 36.135pt=\mathrm{Vol}(B_{T})\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}+\sum_{i=2}^{d^{+}}L_{i}(T)T^{2\frac{\log|\lambda_{i}|}{\log\lambda_{1}}}C_{i}^{n}(g_{1},g_{2})+\mathcal{O}_{n}(T),\end{split}$

where $L_{i}(T)/\log(T)^{s-1}\in\mathcal{O}(1)$ if the size of the Jordan
block associated to the $i^{th}$ eigenvalue of the substitution matrix is $s$.
Moreover, the planar averaged mixing (2) also holds.

######  Remark 2.

Some remarks:

  1. (i)

These results hold independent of whether the tiling models a quasicrystal,
that is, regardless of whether the tiling $\mathcal{T}$ has long range order,
thus applying to systems which do not come from cut-and-project constructions.
In particular, any non-Pisot substitution tiling does not have long range
order [Sol97] but enjoys an asymptotic expansion as in (5).

  2. (ii)

The class of substitution tilings is not the only class for which detailed
asymptotics as in (5) can be given. There is a recently-developed theory of
(globally) random substitution tilings [ST21, Tre21] which produce tilings by
the random composition of different substitution rules, resulting in
repetitive, aperiodic tilings of finite local complexity (and which includes
the self-similar case). The detailed expansion in (5) comes from rates of
deviations of ergodic integrals for the $\mathbb{R}^{2}$ action on the tiling
space for self-similar tilings [BS13], which are generalized in [Tre21] for
random substitution tilings111There are other works which study deviations of
ergodic integrals. However, these make use of cohomology of the tiling space,
which cannot be used here as the functions which we integrate have
discontinuities.. However, the necessary background needed to introduce the
construction of these tilings falls outside the scope of this paper.

  3. (iii)

We remark that our mixing results give new results for _periodic_ Lorentz
gases in that the scatterer configuration can be periodic but the observables
are not. As an example, suppose that all the scatterers in the example in
Figure 2 are modified to be spheres of the same radius (large enough to have
finite horizon). This would give a periodic scatterer configuration. However,
using observables which are $\mathcal{T}$-equivariant with respect to the
half-hex tiling in Figure 1 and which satisfy the conditions of our theorems,
we also obtain planar mixing for this Lorentz gas with periodic scatterer
configuration.

  4. (iv)

The class of pattern-equivariant, locally integrable functions is the largest
class we know for which the detailed expansion (5) holds. However, it is not
the largest class for which the expansion (1) holds. More specifically, the
class of pattern-equivariant functions considered here agrees with the class
of strongly pattern equivariant functions considered in [Kel08]. The mixing
equations (1) and (2) also hold for weakly pattern equivariant functions (see
again [Kel08]), and for certain limits of them. We focus on (strongly)
pattern-equivariant functions here due to their evident relation to the
aperiodic tiling.

  5. (v)

The averages computed using balls $B_{T}$ do not really need to be done with
balls. In fact, any good sequence of averaging sets, such as cubes, Følner, or
Van Hove sequences work just as well. This type of mixing is similar to the
_global-global mixing_ of Lenci [Len10].

  6. (vi)

We do not know whether the Lorentz gases which we consider are _recurrent_ ,
so we do not know whether they belong to the class of aperiodic Lorentz gases
studied by Lenci in [Len03]. Although we study the Lorentz gas via a compact
factor with an invariant probability measure for which Poincaré recurrence
holds, this does not imply the recurrence of the system on $\mathbb{R}^{2}$.

The result on the K property implies the ergodicity of the billiard map
$f:\mathcal{D}\to\mathcal{D}$, which in turn gives a result on ergodicity of
the Lorentz gas flow $\bar{\phi}_{t}$ defined on the tiling space
$\Omega_{\mathcal{T}}$. As such, for any
$\mathcal{T}^{\prime}\in\Omega_{\mathcal{T}}$ we can consider the
corresponding Lorentz gas flow $\phi_{t}^{\mathcal{T}^{\prime}}$ defined on
the plane by the scatterer configuration $\mathcal{S}^{\prime}$ associated to
$\mathcal{T}^{\prime}$.

For any $z\in\partial\mathcal{S}^{\prime}$ and $\theta\in(-\pi/2,\pi/2)$, let
$\tau(z,\theta)\in(0,M]$ denote the free flight time of the Lorentz gas
starting at $z$ with direction $\theta$. We denote by

$\bar{\tau}:=\int_{\mathcal{D}}\tau(\mathcal{T}^{\prime},\theta)\,d\bar{\mu}$

the mean free flight time on $\mathcal{D}$. For a function
$f:\mathbb{R}^{2}\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ define

(6)
$\|f\|_{(1)}:=\sup_{\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)}\sup\left\\{\frac{\int_{\gamma_{\theta}}|f(\cdot,\theta)|}{\mbox{length}(\gamma_{\theta})}:\gamma_{\theta}\mbox{
is a finite length geodesic in $\mathbb{R}^{2}$ in direction
$\theta$}\right\\}$

and for $\mathcal{T}\in\Omega$, the space

(7) $\mathfrak{G}_{\mathcal{T}}:=\left\\{f:f\mbox{ is a locally integrable,
}\mathcal{T}\mbox{-equivariant on $\mathbb{R}^{2}\times(-\pi/2,\pi/2)$ with
}\|f\|_{(1)}<\infty\right\\}.$

For any $g\in\mathfrak{G}_{\mathcal{T}^{\prime}}$ the quantity

(8)
$\bar{G}(z,\theta)=\int_{0}^{\tau(z,\theta)}g\circ\phi^{\mathcal{T}^{\prime}}_{t}(z,\theta)\,dt,$

is well defined, and it is a $\mathcal{T}^{\prime}$-equivariant function on
$\partial\mathcal{S}^{\prime}\times(-\pi/2,\pi/2)$. We prove the following in
Section §7.

######  Theorem 4 (Ergodicity of the Lorentz gas flow).

Let $\mathcal{T}$ be a repetitive, aperiodic tiling of $\mathbb{R}^{2}$ of
finite local complexity, $\mathcal{S}$ be a $\mathcal{T}$-equivariant
collection of scatterers of category A with $C^{3}$ boundary and $\mu$ an
$\mathbb{R}^{2}$-invariant ergodic probability measure on
$\Omega_{\mathcal{T}}$. Let
$h:\Omega\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ be a $L^{1}$ transversally
locally constant function which defines for every
$\mathcal{T}^{\prime}\in\Omega$ a
$g_{\mathcal{T^{\prime}}}\in\mathfrak{G}_{\mathcal{T}^{\prime}}$. For
$\bar{\mu}$-almost every
$(\mathcal{T}^{\prime},\theta)\in\partial\Sigma\times[-\pi/2,\pi/2]$ and $T>0$
we have that

$\frac{1}{T}\int_{0}^{T}g_{\mathcal{T}^{\prime}}\circ\phi^{\mathcal{T}^{\prime}}_{t}(0,\theta)\,dt=\bar{\tau}^{-1}\frac{1}{\mathrm{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}\cap\partial\mathcal{S}^{\prime}}\bar{G}_{\mathcal{T}^{\prime}}\,d\rho\,\cos\theta\,d\theta+o(T),$

as well as

(9)
$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}g_{\mathcal{T}^{\prime}}\circ\phi^{\mathcal{T}^{\prime}}_{t}(0,\theta)\,dt=\bar{\tau}^{-1}\beta(\bar{G}_{\mathcal{T}^{\prime}}).$

We note that the last four items in Remark 2 also apply to this ergodicity
theorem.

This paper is organized as follows. Section 2 covers the necessary background
material from the field of aperiodic tilings and aperiodic point (Delone)
sets. In Section 3 we make rigorous the definition of aperiodic scatterer
configuration under which we work, and we express it both as defined by an
aperiodic tiling or by an aperiodic Delone multiset. We define the aperiodic
Lorentz gas and its compact model, which generalizes the Sinai billiard, and
identify the canonical invariant measure for this system. Section 4 collects
basic facts about stable and unstable manifolds for our compact system, most
of which follow from the theory of Sinai billiards. In Section 5 we prove the
K property for our compact factor. Sections 6 and 7 prove the mixing (Theorems
2 and 3) and ergodicity results (Theorem 4), respectively.

######  Acknowledgements.

We thank F. Rodriguez-Hertz for fruitful discussions on K-systems, as well as
D. Dolgopyat for enlightening discussions on Lorentz gases. We thank the
tornadoes for all the good times.

##  2\. Tilings and tiling spaces

Here we cover the necessary backgroung on tilings. For a more in-depth
treatment, see [Rob04, Sad08, BG13].

A tiling $\mathcal{T}$ of $\mathbb{R}^{d}$ is a covering of $\mathbb{R}^{d}$
by compact sets $t$, called tiles, such that any two tiles of $\mathcal{T}$
may only intersect along their boundaries. The translation
$\varphi_{v}(\mathcal{T})$ of a tiling $\mathcal{T}$ is the tiling obtained by
translating each of the tiles $t$ of $\mathcal{T}$ by the vector
$-v\in\mathbb{R}^{d}$. If there exist compact sets $t_{1},\dots,t_{M}$ such
that every tile $t\in\mathcal{T}$ is translation-equivalent to some $t_{i}$,
then we call the collection $\\{t_{1},\dots,t_{M}\\}$ the prototiles of
$\mathcal{T}$. A union $\mathcal{P}\subset\mathcal{T}$ of a finite number of
tiles of $\mathcal{T}$ is called a patch of $\mathcal{T}$. For any bounded set
$A\subset\mathbb{R}^{d}$ define the patch

$\mathcal{O}^{-}_{\mathcal{T}}(A)=\mbox{ largest patch of $\mathcal{T}$
completely contained in $A$}.$

For two tiles $t,t^{\prime}\in\mathcal{T}$, let $t\sim t^{\prime}$ denote
translation-equivalence, that is, if there exists a $v\in\mathbb{R}^{d}$ such
that $\varphi_{v}(t)=t^{\prime}$, then one writes $t\sim t^{\prime}$.

The types of tilings which will be considered here have the following
properties. They are:

Aperiodic:

If $\mathcal{T}$ is a tiling and $\varphi_{v}(\mathcal{T})=\mathcal{T}$, then
$v=0$.

Repetitive:

For any patch $\mathcal{P}\subset\mathcal{T}$ there exists an $R>0$ such that
every ball of radius at least $R$ around any point in $\mathbb{R}^{d}$
contains a patch which is translation-equivalent to $\mathcal{P}$.

Finite Local Complexity:

For every $R>0$ there is a finite collection of patches
$\\{\mathcal{P}_{i}^{R}\\}$ such that any patch found inside a ball of radius
$R$ is translation equivalent to some $\mathcal{P}_{j}^{R}$.

There is a metric on the set
$\\{\varphi_{v}(\mathcal{T})\\}_{v\in\mathbb{R}^{d}}$ of translates of
$\mathcal{T}$. First, define

$\bar{d}(\mathcal{T},\varphi_{v}(\mathcal{T}))=\inf\left\\{\varepsilon>0:\mathcal{O}^{-}_{\mathcal{T}}(B_{1/\varepsilon}(0))=\mathcal{O}^{-}_{\varphi_{v+w}\mathcal{T}}(B_{1/\varepsilon}(0))\mbox{
for some }\|w\|\leq\varepsilon\right\\},$

and from this define

(10)
$d(\mathcal{T},\varphi_{v}(\mathcal{T}))=\min\left\\{1,\bar{d}(\mathcal{T},\varphi_{v}(\mathcal{T}))\right\\}.$

That (10) is metric is a standard fact in the tilings literature. Denote by

$\Omega_{\mathcal{T}}:=\overline{\\{\varphi_{v}(\mathcal{T}):v\in\mathbb{R}^{d}\\}}$

the metric completion of the set of all translates of $\mathcal{T}$ with
respect to the metric (10). This set is called the tiling space of
$\mathcal{T}$. Note that if $\mathcal{T}$ is repetitive and has finite local
complexity then we have that
$\Omega_{\mathcal{T}^{\prime}}=\Omega_{\mathcal{T}}$ for any other
$\mathcal{T}^{\prime}\in\Omega_{\mathcal{T}}$. By chosing a marked point in
the interior of each prototile $t_{i}$ it can be assumed that every tile in
$\mathcal{T}$ has a marking which is translation equivalent to the marking on
prototiles. The canonical transversal of $\Omega_{\mathcal{T}}$ is the set

(11)
$\mho_{\mathcal{T}}:=\\{\mathcal{T}^{\prime}\in\Omega_{\mathcal{T}}:\mbox{ the
marked point in the tile containing the origin is the origin}\\}.$

Let $\mathcal{P}\subset\mathcal{T}$ be a patch and $t\in\mathcal{P}$ a tile in
the patch. Then

(12)
$\mathcal{C}_{\mathcal{P},t}=\\{\mathcal{T}^{\prime}\in\Omega_{\mathcal{T}}:\mbox{
$\mathcal{P}$ is a patch in $\mathcal{T}^{\prime}$ and the distinguished point
in the tile $t$ is the origin}\\}$

is the $(\mathcal{P},t)$-cylinder set, also called a transversal cylinder set.
For $\varepsilon>0$ the $(\mathcal{P},t,\varepsilon)$-set is defined by

(13)
$\mathcal{C}_{\mathcal{P},t,\varepsilon}:=\bigcup_{\|v\|<\varepsilon}\left\\{\varphi_{v}(\mathcal{T^{\prime}}):\mathcal{T}^{\prime}\in\mathcal{C}_{\mathcal{P},t}\right\\}\subset\Omega_{\mathcal{T}}.$

The tiling space is compact if $\mathcal{T}$ has finite local complexity.
Moreover, if $\mathcal{T}$ is a repetitive tiling of finite local complexity
then the topology of $\Omega_{\mathcal{T}}$ is generated by cylinder sets of
the form $\mathcal{C}_{\mathcal{P},t,\varepsilon}$ for arbitrarily small
$\varepsilon>0$. This gives $\Omega_{\mathcal{T}}$ the local product structure
of $B_{\varepsilon}\times\mathcal{C}$, where $\mathcal{C}$ is a Cantor set and
$B_{\varepsilon}\subset\mathbb{R}^{d}$ is an open ball.

Most of the well-known tilings, such as the Penrose tiling, can be constructed
from a substitution rule, giving the tiling a self-similar hierarchical
structure. This is done as follows.

######  Definition 1.

Given a set of prototiles $t_{1},\dots,t_{M}$ each viewed as a subset of
$\mathbb{R}^{d}$, an expansion and substitution rule is a way of expressing
expanded copies of the prototiles as unions of copies of the prototiles. In
other words, such a rule determines a $\lambda>1$ (the expansion) such that
for any $1\leq i\leq M$,

$\lambda\cdot
t_{i}=\bigcup_{j=1}^{M}\bigcup_{\ell=1}^{r(i,j)}t_{j}+v_{i,j,\ell},$

where $v_{i,j,\ell}\in\mathbb{R}^{d}$ are vectors and $r(i,j)$ is the number
of tiles of type $j$ found in the expanded copy $\lambda t_{i}$ under this
substitution rule.

The matrix $\mathcal{R}$ with $\mathcal{R}_{ij}=r(i,j)$ is called the
substitution matrix for the expansion and substitution. A substitution is
primitive if the substitution matrix is a primitive matrix (i.e.
$\mathcal{R}^{n}$ has all positive entries for all $n$ large enough).

For a tiling $\mathcal{T}$, a patch $\mathcal{P}\subset\mathcal{T}$ and Borel
set $A\subset\mathbb{R}^{d}$, let $L_{\mathcal{T}}(\mathcal{P},A)=\\#\mbox{ of
copies of $\mathcal{P}$ in $A$}$. The asymptotic frequency of a patch
$\mathcal{P}$ in $\mathcal{T}$ is the quantity

(14)
$\mathrm{freq}_{\mathcal{T}}(\mathcal{P}):=\lim_{T\rightarrow\infty}\frac{L_{\mathcal{T}}(\mathcal{P},B_{T})}{{\mathrm{Vol}(B_{T})}}.$

There is a natural action of $\mathbb{R}^{d}$ on $\Omega_{\mathcal{T}}$ by
translation and this action is minimal if $\mathcal{T}$ is repetitive.
Throughout the paper we denote by $\mu$ an invariant Borel probability measure
for the $\mathbb{R}^{d}$-action. By the local product structure of
$\Omega_{\mathcal{T}}$, the measure $\mu$ has a natural product structure
$\mbox{Leb}\times\nu$, where $\nu$ is a transverse measure to the action of
$\mathbb{R}^{d}$ and invariant under holonomy. In this paper we will be
concerned with the cases when $\mu$ is ergodic. Tiling spaces coming from
primitive substitution rules, for example, have a uniquely ergodic action
[Sol97]. In fact, a large class of tiling spaces constructed from random
substitutions also give rise to uniquely ergodic systems. As such, the
assumption that $\mu$ is an ergodic measure is not restrictive.

Since open sets along the totally disconnected transversal
$\mho_{\mathcal{T}}$ are given by cylinder sets of patches, the measure $\nu$
satisfies
$\nu(\mathcal{C}_{\mathcal{P},t})=\mbox{freq}_{\mathcal{T}}(\mathcal{P})$,
where $\mbox{freq}_{\mathcal{T}}(\mathcal{P})$ is the asymptotic frequency of
translation copies of the patch $\mathcal{P}$ in the tiling $\mathcal{T}$. If
there is a unique invariant measure for the $\mathbb{R}^{d}$ action (i.e. the
action is uniquely ergodic), then the patch frequencies [Sol97] are uniform,
so even though frequencies are defined with respect to a tiling, in the
uniquely ergodic case there is no dependence on the choice of a tiling
$\mathcal{T}\in\Omega$, in which case we may write
$\nu(\mathcal{C}_{\mathcal{P},t})=\mbox{freq}(\mathcal{P})$.

####  2.0.1. Approximants and inverse limits

Tiling spaces are inverse limits. That is, there is a family of topological
spaces $\\{\Gamma_{k}\\}_{k\in\mathbb{N}}$ (called the approximants) and maps
$\gamma_{k}=\Gamma_{k}\rightarrow\Gamma_{k-1}$ such that there is a
homeomoprhism

(15)
$\Omega_{\mathcal{T}}\cong\lim_{\leftarrow}(\Gamma_{k},\gamma_{k}):=\left\\{\bar{z}=(z_{1},z_{2},\dots)\in\prod_{k\in\mathbb{N}}\Gamma_{k}:z_{k-1}=\gamma_{k}(z_{k})\right\\}.$

We denote by $\varpi_{k}:\Omega_{\mathcal{T}}\rightarrow\Gamma_{k}$ the
canonical projections to the approximants defined by the inverse limit, and by
$\pi_{\mathcal{T},k}:\mathbb{R}^{d}\rightarrow\Gamma_{k}$ the map
$\pi_{\mathcal{T},k}(\tau)=\varpi_{k}\circ\varphi_{\tau}(\mathcal{T})$. If
$\mathcal{T}$ has finite local complexity, each $\Gamma_{k}$ can be taken to
be a compact space. See [Sad08, Chapter 2] for a full discussion of tiling
spaces as inverse limits. In §3.1 we will use the maps $\pi_{\mathcal{T},0}$
to define aperiodic scatterer configurations.

Figure 3. Approximant for the half-hex tiling.

To make this point of view more palatable, we remark that this point of view
is most elegant when a tiling is constructed from a substitution rule, as
first explained by Anderson and Putnam [AP98]. Figure 3, for example, shows
the approximant $\Gamma_{k}$ for the half-hex tiling in Figure 1 (edges
labeled with the same symbol are identified). Because the tiling is obtained
from a substitution rule (and thus a self-similarity), all approximants
$\Gamma_{k}$ are the same. The maps $\gamma_{k}$ are given by substitution
rule, that is, they subdivide each face of $\Gamma_{k}$ according to the
substitution rule.

###  2.1. Delone sets and multisets

A Delone set $\Lambda\subset\mathbb{R}^{d}$ is a countable subset satisfying
the two conditions

Uniform discreteness:

There exists an $r>0$ such that for any distinct $x,y\in\Lambda$ we have that
$|x-y|>r$.

Relative density:

There is an $R>0$ such that $B_{R}(x)\cap\Lambda\neq\varnothing$ for any
$x\in\mathbb{R}^{d}$.

A Delone multiset is a Delone set of the form
$\Lambda=\bigcup_{i}^{k}\Lambda_{i}$, where each $\Lambda_{i}$ is a Delone set
and the sets $\Lambda_{i}$ are mutually disjoint. We can think of a Delone
multiset as a decorated Delone set. That is each point $x\in\Lambda$ in a
Delone multiset comes ‘‘colored’’ by an index, so $x\in\Lambda$ has color $i$
if $x\in\Lambda_{i}$.

Given a Delone multiset $\Lambda$, a cluster $\mathcal{P}\subset\Lambda$ is a
finite subset of $\Lambda$. It is repetitive if for every cluster
$\mathcal{P}\subset\Lambda$ there exists an $R>0$ such that
$B_{R}(x)\cap\Lambda$ contains a translated copy of $\mathcal{P}$ for any
$x\in\mathbb{R}^{d}$. A Delone multiset has finite local complexity if for
every $R>0$ there exists a finite collection of clusters
$\\{\mathcal{P}_{i}^{R}\\}$ such that any cluster found in a ball of radius
$R$ is translation equivalent to some $\mathcal{P}_{j}^{R}$.

Delone sets and tilings represent two points of view on the same type of
object. Given a tiling $\mathcal{T}$, one can construct a Delone multiset by
taking the center of mass of each tile and coloring them by tile type.
Conversely, from a Delone multiset one can consider the Voronoi tessellation
given by this multiset, giving a tiling of $\mathbb{R}^{d}$. Properties such
as repetitivity, aperiodicity and finite local complexity are preserved by
these types of operations.

Delone multisets are also translated by $\mathbb{R}^{d}$ and one can give the
set of translates of $\Lambda$ a metric analogous to the metric (10) used for
the set of translations of a tiling. The metric completion of this set is the
pattern space $\Omega_{\Lambda}$ of $\Lambda$. Like a tiling space, a pattern
space for a repetitive, aperiodic Delone multiset of finite local complexity
is a foliated metric space, where the leaves correspond to distinct
$\mathbb{R}^{d}$ orbits.

The canonical transversal $\mho_{\Lambda}\subset\Omega_{\Lambda}$ of a pattern
space $\Omega_{\Lambda}$ is the subset

$\mho_{\Lambda}:=\left\\{\Lambda^{\prime}\in\Omega_{\Lambda}:\bar{0}\in\Lambda^{\prime}\right\\}.$

If $\Lambda$ is aperiodic, repetitive and of finite local complexity, then
$\mho_{\Lambda}\subset\Omega_{\Lambda}$ is a Cantor set. Given a cluster
$\mathcal{P}\subset\Lambda$ with $\bar{0}\in\mathcal{P}$, the transversal
cylinder set $\mathcal{C}_{\mathcal{P}}$ is defined in direct analogy to (12).
By shifting such sets by small translations, in analogy to (13), one obtains
cylinder sets $\mathcal{C}_{\mathcal{P},\varepsilon}$ which generate the
topology of the pattern space $\Omega_{\Lambda}$. Just as with tiling spaces,
under the assumptions of aperiodicity, repetitivity and finite local
complexity, the pattern space has local product structure of a Euclidean ball
and a Cantor set.

Given a cluster $\mathcal{P}\subset\Lambda$, the frequency of $\mathcal{P}$ in
$\Lambda$ is computed as in (14). As in tiling spaces, this defines a family
of transverse measures $\nu$ which is invariant under the holonomy of the
$\mathbb{R}^{d}$ action. Furthermore, the measure which is locally the product
$\mathrm{Leb}\times\nu$ is invariant under the $\mathbb{R}^{d}$ action.

###  2.2. Functions

Let $\mathcal{T}$ be a repetitive, aperiodic tiling of $\mathbb{R}^{d}$ of
finite local complexity, and $\Lambda$ a Delone multiset with the same
properties.

######  Definition 2.

For any set $X$, a function $g:\mathbb{R}^{d}\times X\rightarrow\mathbb{R}$ is
$\mathcal{T}$-equivariant (or pattern-equivariant) if there exists an $R>0$
such that if for any $x,y\in\mathbb{R}^{d}$ we have that
$\mathcal{O}^{-}(B_{R}(x))$ (see §2 for definition) is translation equivalent
to $\mathcal{O}^{-}(B_{R}(y))$, then $g(x,z)=g(y,z)$ for any $z\in X$. A
$\Lambda$-equivariant function $g:\mathbb{R}^{d}\times X\rightarrow\mathbb{R}$
is similarly defined for Delone multisets.

In short, these are functions which depend only on the local configuration of
the tiling or Delone multiset used for reference.

######  Definition 3.

Let $\Omega$ be the space associated to either $\mathcal{T}$ or $\Lambda$ and
$X$ any set. A function $h:\Omega\times X\rightarrow\mathbb{R}$ is
transversally locally constant if there exists an $R>0$ such that for any two
$p,p^{\prime}\in\Omega$ we have that $B_{R}(\bar{0})\cap p=B_{R}(\bar{0})\cap
p^{\prime}$ implies $h(p,z)=h(p^{\prime},z)$ for any $z\in X$.

In other words, transversally locally constant functions are locally constant
along transversals. These are the functions with the highest ‘‘regularity’’ in
the direction of the (totally disconnected) transversal. Pattern-equivariant
functions and transversally locally constant functions are related in a nice
way (see [KP06, Proposition 22]).

######  Proposition 1.

For $p\in\Omega$ a function $g:\mathbb{R}^{d}\times X\rightarrow\mathbb{R}$
$p$-equivariant if and only if there is a transversally locally constant
function $h:\Omega\times X\rightarrow\mathbb{R}$ such that

$g(x,z)=h(\varphi_{x}(p),z)$

for all $x\in\mathbb{R}^{d}$ and $z\in X$.

As a direct consequence of Proposition 1 we obtain the following.

######  Corollary 2.

If $\mu$ is an $\mathbb{R}^{d}$-invariant ergodic probability measure on
$\Omega$ and $\kappa$ is any measure on $X$, then for $\mu$\- almost every
$p\in\Omega$ and for every $p$-equivariant function $g:\mathbb{R}^{d}\times
X\rightarrow\mathbb{R}$ we have that

$\lim_{T\to\infty}\frac{1}{\mbox{Vol}(B_{T})}\int_{X}\int_{B_{T}}g(x,z)d\mbox{Vol}(x)d\kappa(z)=\int_{X}\int_{\Omega}h(p^{\prime},z)d\mu(p^{\prime})d\kappa(z),$

where $h:\Omega\times X\rightarrow\mathbb{R}$ is a transversally locally
constant function satisfying $g(x,z)=h(\varphi_{x}(p),z)$ for all
$x\in\mathbb{R}^{d}$ and $z\in X$.

There is a notion of weakly pattern equivariant functions with a similar
characterization, although they will not be featured here. A function
$g:\mathbb{R}^{d}\times X\rightarrow\mathbb{R}$ is weakly $p$-equivariant if
there exists a transversally continuous function $h:\Omega\times
X\rightarrow\mathbb{R}$ such that $g(x,z)=h(\varphi_{x}(p),z)$ for all
$x\in\mathbb{R}^{d}$ and $z\in X$.

##  3\. Lorentz gases

Let $\mathcal{S}\subset\mathbb{R}^{d}$ be a collection of open and convex
topological disks with $C^{3}$ boundaries, called the scatterers. The Lorentz
gas is the system describing the behavior of a particle moving along geodesics
in $\mathbb{R}^{d}\backslash\bar{\mathcal{S}}$ and having elastic collisions
at $\partial\mathcal{S}$. In this paper we restrict our attention to scatterer
configurations of Category A, that is such that the time between any two
collisions with $\partial\mathcal{S}$ is bounded above by some
$M=M(\mathcal{S})>0$ (finite horizon) and that the closures of each two
scatterers are disjoint (no corners).

###  3.1. Aperiodic scatterer configurations

######  Definition 4.

Let $\mathcal{S}\subset\mathbb{R}^{d}$ be a collection of scatterers. We call
$\mathcal{S}$ an aperiodic configuration if there exists no nontrivial
$v\in\mathbb{R}^{d}$ such that $\varphi_{v}(\mathcal{S})=\mathcal{S}$.

In this paper we consider two ways of constructing aperiodic scatterer
configurations, one through aperiodic tilings and the other through aperiodic
Delone multisets. In this subsection we describe both constructions and point
out relations between them.

####  3.1.1. $\mathcal{T}$-equivariant scatterers

Throughout this section, $\mathcal{T}$ is a repetitive, aperiodic tiling of
$\mathbb{R}^{d}$ of finite local complexity.

######  Definition 5.

A collection of $\mathcal{T}$-equivariant scatterers is a scatterer
configuration $\mathcal{S}$ which depends on the local configuration of
$\mathcal{T}$. More precisely, there exists an $R>0$ such that if
$S,S^{\prime}\subset\mathcal{S}$ are connected components (scatterers) and
$x,y$ are their respective centers of mass, then if
$\mathcal{O}_{\mathcal{T}}^{-}(B_{R}(x))=\mathcal{O}_{\mathcal{T}}^{-}(B_{R}(y))+x-y$
then $S=\varphi_{-(x-y)}(S^{\prime})$, that is, the scatterer around $x$ is
translation equivalent to that around $y$.

If $\mathcal{T}$ is a hierarchical tiling, then by the inverse limit
construction in §2.0.1, a collection $\mathcal{S}$ of scatterers is
$\mathcal{T}$-equivariant, if there exists a $k\geq 0$ and an open set
$\mathcal{S}_{k}\subset\Gamma_{k}$ such that
$\mathcal{S}=\pi_{\mathcal{T},k}^{-1}(\mathcal{S}_{k})$. Any such set
$\mathcal{S}_{k}$ which lifts to the scatterers of an aperiodic Lorentz gas is
called the set of model scatterers. Figure 3 shows the model scatterers on the
approximant $\Gamma_{1}$ which gives the pattern equivariant scatterers in
Figure 2.

Figure 4. Approximant and model scatterers.

######  Lemma 1.

Without loss of generality we can assume that $k=0$ for the set of model
scatterers of a set of pattern-equivariant scatterers.

###### Proof.

Suppose that $\mathcal{S}_{k}\subset\Gamma_{k}$ is the set which defines the
$\mathcal{T}$-equivariant collection of scatterers. Let $\Omega_{\mathcal{T}}$
be the tiling space of $\mathcal{T}$ defined as an inverse limit (15). There
is another tiling space $\Omega^{\prime}$ and a homeomorphism of tiling spaces
$\Phi:\Omega_{\mathcal{T}}\rightarrow\Omega^{\prime}$ as follows. For
$\bar{z}=(z_{0},z_{1},z_{2},\dots)\in\Omega_{\mathcal{T}}$ (in inverse limit
coordinates), let $\Phi(z)=(z_{k},z_{k+1},\dots)$, where $k$ is the index of
the approximant where the set of model scatterers is defined. The map $\Phi$
has inverse $\Phi^{-1}=\gamma_{1}\circ\cdots\circ\gamma_{k}$. Thus, if the set
of model scatterers is defined by the $k^{th}$ approximant in
$\Omega_{\mathcal{T}}$, it is defined, up to scaling, by the $0^{th}$
approximant in $\Omega^{\prime}$. ∎

Given this observation, from now on we will assume that our set of scatterers
is defined by a set $\mathcal{S}_{0}\subset\Gamma_{0}$ in the zeroth
approximant.

######  Remark 3.

If $\mathcal{T}$ is a repetitive, aperiodic tiling of finite local complexity
and $\mathcal{S}$ is a $\mathcal{T}$ equivariant collection of scatterers,
then $\mathcal{S}$ is not necessarily aperiodic. This can be observed from
Figures 2 and 4. There is an underlying hexagonal (i.e. periodic) grid in the
half hex tiling. If we were to pick the model scatterers to have the same
constant curvature in Figure 4, they would lift to a periodic configuration in
$\mathbb{R}^{2}$ (see Figure 2).

####  3.1.2. $\Lambda$-equivariant scatterers

Let $\mathcal{S}\subset\mathbb{R}^{d}$ be a scatterer configuration, and
$\mathfrak{C}(\mathcal{S})$ the set of connected components of $\mathcal{S}$.
Let $\overline{\mathfrak{C}(\mathcal{S})}=\mathfrak{C}(\mathcal{S})/\sim$,
where $S\sim S^{\prime}$ if and only if $S=\varphi_{v}(S^{\prime})$ for some
$v\in\mathbb{R}^{d}$. The set $\overline{\mathfrak{C}(\mathcal{S})}$ records
the geometrically different types of scatterers found in the collection
$\mathcal{S}$. For each $c\in\overline{\mathfrak{C}(\mathcal{S})}$ we set

$\Lambda_{c}^{\mathcal{S}}=\bigcup_{S\in c}\\{\mbox{center of mass of
}S\\}\hskip 28.90755pt\mbox{ and define }\hskip
28.90755pt\Lambda_{\mathcal{S}}=\bigcup_{c\in\overline{\mathfrak{C}(\mathcal{S})}}\Lambda_{c}^{\mathcal{S}},$

which may or may not be a Delone multiset.

######  Definition 6.

Let $\Lambda=\bigcup_{i\in I}\Lambda_{i}$ be a Delone multiset. A scatterer
configuration $\mathcal{S}$ is $\Lambda$-equivariant if there are $|I|$
distinct $C^{3}$ convex (topological) $d-1$ spheres $\\{S_{i}\\}$ such that
$\Lambda_{i}=\Lambda_{i}^{\mathcal{S}}$ and $\Lambda_{\mathcal{S}}=\Lambda$.

######  Definition 7.

We say that a configuration of scatterers $\mathcal{S}$ is repetitive, has
finite local complexity, and uniform patch frequency, respectively, if and
only if $\Lambda_{\mathcal{S}}$ forms a Delone multiset with those properties.

######  Lemma 2.

If $\mathcal{S}$ is a repetitive, aperiodic scatterer configuration with
finite local complexity and uniform patch frequency then there is a
repetitive, aperiodic tiling $\mathcal{T}$ of finite local complexity and
uniform patch frequency such that $\mathcal{S}$ is a $\mathcal{T}$-equivariant
scatterer configuration.

###### Proof.

Consider the Delone multiset $\Lambda_{\mathcal{S}}$ and consider the Voronoi
tessellation $\mathcal{T}_{\mathcal{S}}$ defined by $\Lambda_{\mathcal{S}}$.
Then the scatterer configuration depends inside of which tile type its center
of mass is, so it is $\mathcal{T}_{\mathcal{S}}$-equivariant. ∎

Let $\mathcal{S}\subset\mathbb{R}^{d}$ be a repetitive, aperiodic scatterer
configuration of finite local complexity and let $\Omega=\Omega_{\mathcal{S}}$
be the pattern space associated to $\mathcal{S}$ (or to the corresponding
Delone multiset $\Lambda_{\mathcal{S}}$). The set of scatterers $\mathcal{S}$
can be seen as a particular subset of $\Omega$ as follows. Consider the set of
translates $\varphi_{v}(\mathcal{S})$ of the scatterer configuration
$\mathcal{S}$ such that the origin is contained in a scatterer, that is,

$\Sigma_{\mathcal{S}}^{\prime}:=\\{v\in\mathbb{R}^{d}:\bar{0}\in\varphi_{v}(\mathcal{S})\\}.$

This defines a subset $\Sigma_{\mathcal{S}}$ of the orbit of
$\Lambda_{\mathcal{S}}$ in $\Omega$ by

$\Sigma_{\mathcal{S}}=\bigcup_{v\in\Sigma_{\mathcal{S}}^{\prime}}\varphi_{v}(\Lambda_{\mathcal{S}})$

whose closure is the set
$\Sigma:=\overline{\Sigma_{\mathcal{S}}}\subset\Omega$. The set $\Sigma$ now
defines an aperiodic configuration of $\Lambda$-equivariant scatterers for any
Delone multiset $\Lambda\in\Omega$, not just in the $\mathbb{R}^{d}$ orbit of
$\Lambda_{\mathcal{S}}$. We denote by $\partial\Sigma$ the corresponding set
of boundaries of scatterers in $\Sigma$. This set has the following local
product structure

(16)
$\partial\Sigma=\bigsqcup_{i\in\overline{\mathfrak{C}(\mathcal{S})}}\partial
S_{i}\times\mathcal{C}_{i},$

where $\\{S_{i}\\}$ is a finite collection of convex topological $d-1$-spheres
(model scatterers), and $\mathcal{C}_{i}$ is some Cantor set. As such, any
$\Lambda\in\partial\Sigma$ corresponds to an aperiodic scatterer configuration
$\mathcal{S}_{\Lambda}\subset\mathbb{R}^{d}$ with the property that
$\bar{0}\in\partial\mathcal{S}_{\Lambda}$.

Let us explain how one can think of the Cantor sets appearing in (16). Let
$\Lambda\in\mho\subset\Omega$ be a Delone multiset which contains the origin
and $R^{\prime}$ the radius for the relative density of $\Lambda$. As such,
its associated scatterer configuration $\mathcal{S}_{\Lambda}$ not only covers
the origin, but the origin is the center of mass of the scatterer which covers
the origin. Suppose this is a scatterer of type $i$, so
$\Lambda\in\mathcal{C}_{i}$. If $\Lambda^{\prime}\in\mho$ is any other Delone
multiset such that its scatterer centered at the origin is of type $i$ then it
also belongs to $\mathcal{C}_{i}$. By finite local complexity, there are
finitely many clusters in a ball of radius $4R^{\prime}$ which appear in all
the Delone multisets of $\Omega$. As such, if
$\Lambda^{\prime}\in\mathcal{C}_{i}\subset\mho$, then there are finitely
configurations in which the scatterer at the origin can be surrounded by
scatterers intersecting $B_{4R^{\prime}}(0)$. This partitions
$\mathcal{C}_{i}$ into finitely many subsets, one for each local configuration
around the origin up to a distance of $4R^{\prime}$, that is, depending on the
scatterer configurations around a ball of radius $4R^{\prime}$. Each of these
elemens of the partition are further partitioned by the finitely many possible
ways in which one finds scatterer configurations in balls of radius
$8R^{\prime}$ around the origin, and so on. This choice of finitely many
options at every stage makes $\mathcal{C}_{i}$ a Cantor set.

If $\mathcal{S}$ is $\mathcal{T}$-equivariant, with $\mathcal{T}$ a
hierarchical, aperiodic, repetitive tiling of finite local complexity, then
$\partial\Sigma\subset\Omega_{\mathcal{T}}$ is obtained as follows. Let
$\mathcal{S}_{0}\subset\Gamma_{0}$ the open set on the approximant that lifts
to the collection of scatterers on $\mathbb{R}^{d}$ (Lemma 1) and let
$\Sigma=\varpi^{-1}_{0}(\mathcal{S}_{0})$ be the lift of the model scatterers
to the tiling space. Then
$\partial\Sigma=\varpi^{-1}_{0}(\partial\mathcal{S}_{0})$. In addition,
$\mathcal{T}\in\partial\Sigma$ if $\mathcal{T}$ is a tiling such that the
origin lies on the boundary of a scatterer.

###  3.2. The compact model

As in the case of periodic scatterer configurations for the Lorentz gas, it
will be easier to work with a factor system defined on a compact set. We
describe the construction in this section.

Let $\Omega$ be a tiling space (or pattern space) corresponding to an
aperiodic, repetitive tiling (or Delone multiset) with finite local
complexity, and let $\Sigma\subset\Omega$ be a subset which defines for any
$p\in\Omega$ an aperiodic $p$-equivariant collection of $C^{3}$ convex
scatterers $\mathcal{S}_{p}\subset\mathbb{R}^{d}$ of category A. Points on
boundaries $\partial\mathcal{S}_{p}$ of scatterers will be given directions
away from scatterers, and this is parametrized by the upper half of the $d-1$
unit sphere $S^{d-1}_{+}\subset T_{p}\mathbb{R}^{d}$.

For $(p,v_{0})\in(\Omega\backslash\bar{\Sigma}\times
S^{d-1})~{}\cup~{}(\partial\Sigma\times S^{d-1}_{+})$ define

$\tau_{0}:=\min\left\\{r>0:\varphi_{rv_{0}}(p)\in\partial\Sigma\right\\}.$

Let $n_{1}\in S^{d-1}$ be the unit normal vector at the point
$\tau_{0}v_{0}\in\partial\mathcal{S}_{p}$ (equivalently, the unit normal
vector to the scatterer at the origin in $\varphi_{\tau_{0}v_{0}}(p)$). We
then define $v_{1}:=v_{0}-2\langle v_{0},n_{1}\rangle n_{1}$. Recursively,
define for $k\geq 1$

$V_{k-1}:=\sum_{i=0}^{k-1}\tau_{i}v_{i},\hskip
25.29494pt\tau_{k}:=\min\left\\{r>0:\varphi_{rv_{k}}\circ\varphi_{V_{k-1}}(p)\in\partial\Sigma\right\\},$

$n_{k+1}\in S^{d-1}$ to be the unit normal vector at the point
$V_{k}\in\partial\mathcal{S}_{p}$ (equivalently, the tangentially normal at
$\varphi_{\tau_{k}v_{k}}\circ\varphi_{V_{k-1}}(p)\in\partial\Sigma),$ and

$v_{k+1}=v_{k}-2\langle v_{k},n_{k+1}\rangle n_{k+1}.$

Since our starting points were arbitrary, any
$(p,v_{0})\in(\Omega\backslash\bar{\Sigma}\times
S^{d-1})~{}\cup~{}(\partial\Sigma\times S^{d-1}_{+})$ defines sequences
$\\{(\tau_{i},v_{i})\\}$ with $|\tau_{i+1}-\tau_{i}|\leq M$ by finite horizon
(see the beginning of Section 3).

For $k\in\mathbb{N}$ denote $L_{k}:=\sum_{i=0}^{k}\tau_{i}$. For any pair
$(p,v_{0})\in(\Omega\backslash\bar{\Sigma}\times
S^{d-1})~{}\cup~{}(\partial\Sigma\times S^{d-1}_{+})$, the Lorentz gas
trajectory for the scatterer configuration $\mathcal{S}_{p}$ starting from the
origin and in direction $v_{0}$ is given for all $t\geq 0$ by

$\phi^{p}_{t}(\bar{0},v_{0})=\left(V_{k}+\left(t-L_{k}\right)v_{k+1},v_{k+1}\right)=:(\phi^{p,v_{0}}_{t}(\bar{0}),v_{k+1}),$

where $k$ is such that $L_{k}\leq t<L_{k+1}.$ More generally, for
$\omega\in\mathbb{R}^{d}\setminus\mathcal{S}_{p}$ we define
$\phi^{p}_{t}(\omega,v_{0})=\phi^{\varphi_{\omega}(p)}_{t}(\bar{0},v_{0}).$ As
a system evolving in the tiling space (or pattern space), the system is
defined by

$\bar{\phi}_{t}(p,v_{0})=\left(\varphi_{\phi^{p,v_{0}}_{t}(\bar{0})}(p),v_{k+1}\right)$

for all $t\geq 0$. Note that this implies that
$\bar{\phi}_{L_{k}}(p,v_{0})\in\partial\Sigma$ for all $k\in\mathbb{N}$.

Since the collisions at $\partial\mathcal{S}_{p}$ serve as a Poincaré section
of the flow

$\phi^{p}_{t}:(\mathbb{R}^{d}\setminus\overline{\mathcal{S}_{p}}\times
S^{d-1})\cup(\partial\mathcal{S}_{p}\times
S^{d-1}_{+})\to(\mathbb{R}^{d}\setminus\overline{\mathcal{S}_{p}}\times
S^{d-1})\cup(\partial\mathcal{S}_{p}\times S^{d-1}_{+}),$

the set $\partial\Sigma\times S^{d-1}_{+}$ serves as a Poincaré section of the
flow

$\bar{\phi}_{t}:(\Omega\backslash\bar{\Sigma}\times
S^{d-1})~{}\cup~{}(\partial\Sigma\times
S^{d-1}_{+})\to(\Omega\backslash\bar{\Sigma}\times
S^{d-1})~{}\cup~{}(\partial\Sigma\times S^{d-1}_{+}).$

Denote by $f:\mathcal{D}\rightarrow\mathcal{D}$ the Poincaré map, with
$\mathcal{D}:=\partial\Sigma\times S^{d-1}_{+}$ called the collision space.
Using (16), the collision space inherits a local product structure from
$\Omega$ as follows. A neighborhood in $\partial\Sigma\times S^{d-1}_{+}$ is
defined by

  1. (i)

a neighborhood in $\partial S_{i}$ for some
$i\in\overline{\mathfrak{C}(\mathcal{S})}$ describing the translation class of
the scatterer where the collision is taking place (a small subset homeomorphic
to a disk of dimension $d-1$);

  2. (ii)

a clopen set in $\mathcal{C}_{i}$ describing a local configuration of
scatterers around the a collision point, that is, a cylinder set defined by a
cluster $\mathcal{P}$ of the Delone multiset $\Lambda\in\partial\Sigma$ with
$\bar{0}\in\mathcal{P}\subset\Lambda$;

  3. (iii)

and a neighborhood in $S^{d-1}_{+}$ homeomorphic to a disc of dimension $d-1$
describing the direction in which the trajectory goes after the collision.

Thus the collision set can be given coordinates from the set

(17) $\partial\Sigma\times S^{d-1}_{+}=\left(\bigsqcup_{i}\partial
S_{i}\times\mathcal{C}_{i}\right)\times S^{d-1}_{+}.$

In other words, any point in $\mathcal{D}$ can be written in local coordinates
as $(r,c,v)$ where $r\in\partial S_{i}$, $c\in\mathcal{C}_{i}$ belongs to a
transverse cylinder set of the pattern space $\Omega$, and $v\in S^{d-1}_{+}$
is a direction. We note that each path connected component of this space is
isometric to $\partial S_{i}\times S^{d-1}_{+}$ for some $i$. We will refer to
these components as the connected components of $\mathcal{D}$. For
$x\in\mathcal{D}$ we will denote the connected component containing $x$ by
$\mathcal{L}_{x}$. A set $A\subset\mathcal{D}$ is leafwise open if
$A\subset\mathcal{L}$ for some connected component
$\mathcal{L}\subset\mathcal{D}$ and open in the subspace topology on
$\mathcal{L}$ inherited from the topology of $\mathcal{D}$. A leafwise open
neighborhood of $x$ is the intersection of an open neighborhood of $x$ in
$\mathcal{D}$ and the connected component containing $x$.

######  Definition 8.

A canonical cylinder set of the collision space is a subset
$C_{I,\mathcal{P},D}\subset\partial\Sigma\times S^{d-1}_{+}$ of the collision
space expressed in the local coordinates (17) where $I\subset\partial S_{i}$
is a small ball, $\mathcal{P}$ is a patch/cluster around the origin having a
point of type $i$ at the origin, and a small ball of direction vectors in
$D\subset S^{d-1}_{+}$.

######  Remark 4.

Some remarks:

  1. (i)

Since patches define subsets of $\mho$, a canonical cylinder set is partly
defined by a subset $C\subset\mho$.

  2. (ii)

The topology of the collision space $\partial\Sigma\times S^{d-1}_{+}$ is
generated by canonical cylinder sets. Moreover, by compactness, for any
$\varepsilon>0$ there exist finitely many open balls
$\mathfrak{I}_{\varepsilon}=\\{I_{i}\\}$, $I_{i}\subset\mathcal{S}_{0}$ of
radius $\varepsilon$, patches
$\mathfrak{P}_{\varepsilon}=\\{\mathcal{P}_{j}\\}$ of radius
$2\varepsilon^{-1}$, and finitely many open balls
$\mathfrak{D}_{\varepsilon}=\\{D_{k}\\}$ of $S^{d-1}_{+}$ of radius
$\varepsilon$ such that
$\\{C_{I,\mathcal{P},D}~{}|~{}I\in\mathfrak{I}_{\varepsilon},~{}\mathcal{P}\in\mathfrak{P}_{\varepsilon},~{}D\in\mathfrak{D}_{\varepsilon}\\}$
covers the collision space $\partial\Sigma\times S^{d-1}_{+}$.

The pattern space $\Omega$ is a foliated space and, as such, so is the set
$\Omega\times S^{d-1}_{+}$ by taking the leaves to be of the form
$\mathcal{L}\times S^{d-1}_{+}$ for any leaf $\mathcal{L}$ in the foliation of
$\Omega$. As such, the collision space $\mathcal{D}$ inherits a foliated
structure, where the leaves are the connected components in (17). This allows
us to define (leafwise) tangential Jacobians as follows: for
$x=(r,c,v)\in\mathcal{D}$ which does not lie on a discontinuity set of $f$,
the tangential Jacobian $D_{x}^{t}f$ as a map from a leafwise open
neighborhood of $x$ to a leafwise open neighborhood of $f(x)$.

###  3.3. Planar aperiodic Lorentz gases

We now focus on the planar case, when $d=2$. In this case, the prefered
coordinates in (17) are parametrized by $(\rho,c,\theta)$, where
$\rho\in\partial S_{i}$ is an arclength parameter, $c\in\mathcal{C}_{i}$ is an
element of the zero dimensional transverse set, and $\theta\in[-\pi/2,\pi/2]$
is the angle of collision at $\rho$, chosen so that the unit outward normal
corresponds to $\theta=0$. We will denote the corresponding projection maps
$\pi_{i}$ from (17), noting that the last coordinate takes values in
$[-\pi/2,\pi/2]$. Denote by $\Xi_{1}$ the set of singularities of the map $f$.
These correspond to tangential collisions: $\bar{p}\in\Xi_{1}$ if and only if
$\pi_{3}(f(\bar{p}))=\pm\pi/2$. These are ‘‘Cantorized’’ unions of compact
smooth curves. In other words, there is a finite partition of the Cantor sets
$\mathcal{C}_{i}$ in (17)

$\mathcal{D}=\left(\bigsqcup_{i}\partial
S_{i}\times\bigsqcup_{j=1}^{s(i,1)}\mathcal{C}_{i,j}^{1}\right)\times[-\pi/2,\pi/2]$

such that there are finitely many curves $\gamma_{i,j}\subset\partial
S_{i}\times[-\pi/2,\pi/2]$ with $\Xi_{1}$ homeomorphic to

(18) $\bigsqcup_{i,j}\gamma_{i,j}\times\mathcal{C}_{i,j}^{1}.$

Likewise denote by $\Xi_{-1}$ the set of singularities of $f^{-1}$ and
recursively,

(19) $\Xi_{\pm(n+1)}=\Xi_{\pm n}\cup f^{\mp 1}(\Xi_{\pm n}),\hskip
18.06749pt\Xi^{\pm}=\bigcup_{n\geq 0}\Xi_{\pm n},\hskip 18.06749pt\mbox{ and
}\hskip 18.06749pt\Xi=\Xi^{+}\cup\Xi^{-}.$

Likewise for each $n\in\mathbb{N}$ there is a finite partition
$\\{\mathcal{C}_{i,j}^{n}\\}$ of the union of the Cantor sets
$\\{\mathcal{C}_{i}\\}$ such that $\Xi_{\pm n}^{\pm}$ can be expressed as the
cantorized union of curves as in (18).

Let $\mu$ be a Borel probability measure on the tiling (or pattern) space
$\Omega$ which is ergodic with respect to the $\mathbb{R}^{2}$ action and
recall its local product structure $\mathrm{Leb}\times\nu$, where $\nu$ is the
measure on the zero dimensional transversal defined by frequencies of patches
(or clusters). Since the boundaries of the scatterers are sufficiently smooth
($C^{3}$), there is a naturally induced measure on $\partial
S_{i}\times\mathcal{C}_{i}\times[-\pi/2,\pi/2]$ which has the local product
structure of $\mathrm{Leb}\times\nu\times\mathrm{Leb}$. The sets of
singularities $\Xi^{\pm}$ have zero $\mathrm{Leb}\times\nu\times\mathrm{Leb}$
measure. Consider the measure $\bar{\mu}$ on $\mathcal{D}$ expressed in local
coordinates as $\bar{\mu}:=\cos\theta\,d\rho\,d\nu\,d\theta$. The extended set
of collisions with eventual singularities $\Xi$ has zero $\bar{\mu}$ measure.

######  Proposition 2.

The map
$f:\partial\Sigma\times[-\pi/2,\pi/2]\longrightarrow\partial\Sigma\times[-\pi/2,\pi/2]$
preserves the measure $\bar{\mu}$.

###### Proof.

Let $\pi_{i}$, $i=1,2,3$, be the different projections from the coordinates in
(17). Let $A$ be a canonical cylinder set of the form
$A=U\times\mathcal{C}_{i}\times I$ with the property that $A$ does not
intersect the set of discontinuities of $f$. The set $A$ parametrizes an open
set of trajectories starting at a scatterer of type $i$ which travel together
before hitting another scatterer. By finite local complexity, there are
finitely many families of trajectories defined by $A$. Each of these different
families of trajectories is determined by a two-point cluster
$\mathcal{P}_{A,j}$, where one point is of type $i$, being the center of mass
of a scatterer of type $i$ from which the trajectory is emanating and located
at $\bar{0}$, and where the other point in the cluster is determined by the
center of mass and type of the scatterer hit in the first collision. We index
all possible families of trajectories by $\mathfrak{I}(A)$ and stress that
this is a finite set by finite local complexity. We then have that

$A=\bigsqcup_{j\in\mathfrak{I}(A)}A_{j}\text{ with
}\sum_{j\in\mathfrak{I}(A)}\mathrm{freq}(\mathcal{P}_{A,j})=\nu(\mathcal{C}_{i}).$

We can now show the invariance of the measure:

(20)
$\begin{split}\bar{\mu}(f(A))&=\sum_{j\in\mathfrak{I}(A)}\bar{\mu}(f(A_{j}))=\sum_{j\in\mathfrak{I}(A_{j})}\iiint_{f(A_{j})}\cos\theta\,d\theta\,d\nu\,d\rho\\\
&=\sum_{j\in\mathfrak{I}(A_{j})}\mathrm{freq}(\mathcal{P}_{A,j})\int_{\pi_{1}(f(A_{j}))}\int_{\pi_{3}(f(A_{j}))}\cos\theta\,d\theta\,d\rho\\\
&=\sum_{j\in\mathfrak{I}(A_{j})}\mathrm{freq}(\mathcal{P}_{A,j})\int_{\pi_{1}(A_{j})}\int_{\pi_{3}(A_{j})}\cos\theta\,d\theta\,d\rho\\\
&=\sum_{j\in\mathfrak{I}(A_{j})}\iiint_{A_{j}}\cos\theta\,d\theta\,d\nu\,d\rho=\int_{A}\,d\bar{\mu}=\bar{\mu}(A),\end{split}$

where we have used the preservation of the leafwise measure [CM06, Lemma 2.35]
to obtain the fourth equality. For smaller canonical cylinder sets $A$ (i.e.
ones where we use a much smaller set $C\subset\mathcal{C}_{i}$) we do not
actually need to break up the set $A$ by clusters and a simpler version of
(20) will hold, showing the invariance of the measure $\bar{\mu}$. ∎

##  4\. Stable and unstable manifolds

In this section we collect results about the properties of stable and unstable
manifolds. Most of the results follow from the theory of Sinai billiards, so
we frequently refer to a standard reference for these objects [CM06]. The
reason that most results follow from the theory of Sinai billiards can be
briefly summarized as follows. We consider systems with finite local
complexity, therefore having finitely many scatterer types, which results in
uniform (upper and lower) bounds on their curvature. Another fact that follows
from finite local complexity is finitely many possible local configurations of
scatterers, which gives uniform lower bound on the time between collisions. In
addition, we assume that our system has finite horizon, giving an upper bound
on the time between collisions. Those facts lead to an existence of stable and
unstable cones on the (leafwise) tangent bundle (see Definition 9) of the
collision space $\mathcal{D}=\partial\Sigma\times[-\pi/2,\pi/2]$. As in the
case of periodic Sinai Billiards, those cones are strictly invariant and
exhibit uniform contraction and expansion respectively. This implies that
leaves of the foliation of $\mathcal{D}$ given by the orbits of the
$\mathbb{R}^{2}$-action admit local stable and unstable manifolds at Leb-
almost every point. In addition, leafwise analysis of stable/unstable curves,
stable/unstable manifolds, stable/unstable homogeneus manifolds, as well as
leafwise analysis of singularity curves is identical to the periodic case.
Consequently, all properties of periodic billiards established in [CM06,
Chapter 4 and Chapter 5] have their leafwise versions in our setting. This
includes results regarding lengths and regularity of stable/unstable
(homogeneus) manifolds, results concerning the regularity of the holonomy map,
growth lemmas, and the Fundamental theorem of the theory of dispersing
billiards. In this section we summarize some of the main facts, skipping many
of them. We emphasize that arguments in our paper rely on two major results:
the absolute continuity of the holonomy map, as well as the Fundamental
theorem of the theory of dispersing billiards.

Let $\mathcal{T}$ be a repetitive tiling of $\mathbb{R}^{2}$ of finite local
complexity. Consider a $\mathcal{T}$-equivariant collection of $C^{3}$
dispersing scatterers (that is scatterers of positive curvature) of category
A.

######  Definition 9.

Let $T\partial\Sigma$ denote the sub-bundle of the (leafwise) tangent bundle
$T\Omega_{\mathcal{T}}$ corresponding to the restriction of
$T\Omega_{\mathcal{T}}$ to $\partial\Sigma$. We define the  (leafwise) tangent
bundle of the collision space $\mathcal{D}=\partial\Sigma\times[-\pi/2,\pi/2]$
to be

$T\mathcal{D}:=T\partial\Sigma\times[-\pi/2,\pi/2].$

Define the stable/unstable cones at $p\in\mathcal{D}$ to be

$\mathcal{C}_{p}^{u}=\\{(dr,d\phi)\in T_{p}\mathcal{D}:\mathcal{K}_{p}\leq
d\phi/dr\leq\mathcal{K}_{p}+\cos\phi/\tau_{-1}\\},$

where $\mathcal{K}_{p}$ is the curvature of the scatterer at $p$, and
$\tau_{-1}$ is the time between the colision at $f^{-1}(p)$ and $p$, and

$\mathcal{C}_{p}^{s}=\\{(dr,d\phi)\in
T_{p}\mathcal{D}:-\mathcal{K}_{p}-\cos\phi/\tau_{1}\leq
d\phi/dr\leq-\mathcal{K}_{p}\\}.$

Both families of cones are strictly invariant [CM06, §4.4]:

(21)
$D^{t}_{p}f(\mathcal{C}_{p}^{u})\subset\mathrm{int}\,\mathcal{C}_{f(p)}^{u}\cap\\{0\\}\hskip
43.36243pt\mbox{ and }\hskip
43.36243ptD^{t}_{p}f^{-1}(\mathcal{C}_{p}^{s})\subset\mathrm{int}\,\mathcal{C}_{f^{-1}(p)}^{s}\cap\\{0\\}.$

In addition, setting $\Lambda:=1+2\tau_{min}\mathcal{K}_{\min}>1,$ where
$\tau_{min}$ denotes the shortest time between collisions and
$\mathcal{K}_{\min}$ is the lower bound on the curvature of a scatterer, we
obtain that [CM06, §4.4]

$||D^{t}_{p}f^{n}(dx^{u})||\geq\hat{c}\Lambda^{n}||dx^{u}||\hskip
43.36243pt\mbox{ and }\hskip
43.36243pt||D^{t}_{p}f^{-n}(dx^{s})||\geq\hat{c}\Lambda^{n}||dx^{s}||$

for any $p\in\mathcal{D}$, some $\hat{c}>0$, any
$dx^{u}=(dr^{u},d\phi^{u})\in\mathcal{C}_{p}^{u}$ and any
$dx^{s}=(dr^{s},d\phi^{s})\in\mathcal{C}_{p}^{s}$ thus establishing uniform
leafwise hyperbolicity of the map $f:\mathcal{D}\rightarrow\mathcal{D}$.
However, the transverse direction does not exhibit any hyperbolic behavior.
Indeed, if one considers a periodic point $p=(r,c,\phi)$ it is immediate that
any point $p^{\prime}=(r,c^{\prime},\phi)$ for $c^{\prime}\in\mathcal{C}_{0}$
close enough to $c\in\mathcal{C}_{0}$ will also be periodic. In fact, the same
argument shows that any _bounded_ orbit of the aperiodic Lorenz gas comes in a
Cantorized family of bounded orbits. The following Lemma makes precise how the
compact model $f$ is a partially hyperbolic system for any orbit. First, let
$B_{\varepsilon}^{\mho}(x)$ be the ball of radius $\varepsilon$ containing $x$
along the transversal direction. That is, for $x=(r,c,\phi)$

$B^{\mho}_{\varepsilon}(x)=\\{z=(r^{\prime},c^{\prime},\phi^{\prime})\in\mathcal{D}:r^{\prime}=r,\,\phi^{\prime}=\phi,\;d(x,z)<\varepsilon\\}.$

Note that this defines a patch $\mathcal{P}$ and tile $t$ such that
$B_{\varepsilon}^{\mho}(x)=\mathcal{C}_{\mathcal{P},t}$.

######  Lemma 3.

There exists a $\tau>0$ such that for all $\varepsilon\in(0,\tau)$,

$f^{k}\left(B^{\mho}_{\varepsilon}(x)\right)\subset
B^{\mho}_{2\varepsilon}\left(f^{k}(x)\right)$

for all $0\leq k<\lfloor\tau/\varepsilon\rfloor/2$.

###### Proof.

By finite horizon there is a $M>0$ such that the time/distance between two
collissions are bounded above by $M$. Likewise, since the scatterers are
$\mathcal{T}$-equivariant, there is an $R$ so that if two tilings agree on a
ball of radius $R$ then their scatterers around the origin are translation-
equivalent. Let $\varepsilon<\tau:=1/4R$.

If $z\in B_{\varepsilon}^{\mho}(x)$, then the patches around the corresponding
tilings contained in a ball of radius $\varepsilon^{-1}$ are identical, and so
the first $\lfloor(\varepsilon^{-1}-2R)/M\rfloor$ collisions will be
identical. That is, the path followed by the billiard trajectory after
$\lfloor(\varepsilon^{-1}-2R)/M\rfloor$ collisions starting at the initial
conditions defined by $x$ will be indistinguishable from those defined by $z$.
Thus, after $k<(\lfloor(\varepsilon^{-1}-2R)/M)\rfloor/2$ collisions, the
patches of radius $\varepsilon^{-1}/2$ around the origin of $f^{k}(x)$ and
$f^{k}(z)$ will be the same, meaning that $f^{k}(z)\in
B_{2\varepsilon}(f^{k}(x))$, proving the claim. Note that we have proved that
for all $k$ large enough,
$f^{i}\left(B^{\mho}_{\frac{1}{kM-2R}}(x)\right)\subset
B^{\mho}_{\frac{2}{kM-2R}}\left(f^{i}(x)\right)$ for all $0\leq i<k/2$. ∎

By the local structure of $\mathcal{D}$, any curve $W\subset\mathcal{D}$ of
finite length is necessarily contained path-connected components of
$\mathcal{D}$, that is, $W$ is contained in a leaf of the foliated space
$\mathcal{D}$. A curve $W$ is $C^{m}$ if it is leafwise-$C^{m}$. A $C^{m}$
curve $W$ is unstable if for any $p\in W$ the leafwise tangent line to $W$ at
$p$ is contained in $\mathcal{C}^{u}_{p}$. A $C^{m}$ curve $W$ is stable if
for any $p\in W$ the leafwise tangent line to $W$ at $p$ is contained in
$\mathcal{C}^{s}_{p}$.

######  Definition 10.

A smooth curve $W\subset\mathcal{D}$ is an unstable manifold if $f^{n}$ is
smooth on $W$ for all $n<0$ and $|f^{-n}(W)|\rightarrow 0$ as
$n\rightarrow\infty$. A smooth curve $W\subset\mathcal{D}$ is a stable
manifold if $f^{n}$ is smooth on $W$ for all $n>0$ and $|f^{n}(W)|\rightarrow
0$ as $n\rightarrow\infty$.

The sets $\Xi^{+}$ and $\Xi^{-}$ of discontinuities defined in (19) are dense
in $\mathcal{D}$; the proof from the periodic case extends to the setting here
[CM06, Lemma 4.55]. The set $\Xi^{+}\subset\mathcal{D}$ is the union of smooth
unstable curves while $\Xi^{-}$ is the union of smooth stable curves. For each
$n\geq 0$, the set $\Xi_{n}$ consists of a family of transversally locally
constant curves. More precisely, consider the set $\Xi_{1}$, the set of
discontinuities of the map $f$, and a point $p\in\Xi_{1}$. Denote by
$\Xi^{p}_{1}$ the path connected component of $\Xi_{1}$ containing $p$. The
point $p$ corresponds to a type of billiard trajectory which leaves a
scatterer and has a tangential collision with another scatterer in time less
than $M$. If the tiling is repetitive, then the patch allowing a translation-
equivalent type of tangential collision is found within a bounded distance of
every point. Each of these patches correspond to grazing collisions of the
same type, and they correspond to other connected components of $\Xi_{1}$
which are all contained in a transverse neighborhood of $\Xi_{1}^{p}$. More
generally, for $n>1$, the set $\Xi_{n}$ the union of finitely many
transversally-locally-constant curves. As $|n|$ increases, the transverse
neighborhood containing path connected components of
$\Xi_{n}\backslash\Xi_{n-1}$ decreases in size. Another way of stating this is
saying that if $p=(r,c,\phi)\in\Xi_{n}$ then
$p^{\prime}=(r,c^{\prime},\phi)\in\Xi_{n}$ for all $c^{\prime}$ close enough
to $c$ in $\mathcal{C}_{0}$.

For $k\geq k_{0}$ (some large constant) and a transverse coordinate
$c\in\mathcal{C}_{0}$, define the homogeneity strips at $c$ to be

(22)
$\begin{split}\mathbb{H}_{k}^{c}:=\left\\{(r,c,\phi):\frac{\pi}{2}-k^{-2}<\phi<\frac{\pi}{2}-(k+1)^{-2}\right\\}\hskip
36.135pt&\mbox{ with }\hskip
18.06749pt\mathbb{H}_{k}:=\bigcup_{c\in\mathcal{C}_{0}}\mathbb{H}_{k}^{c},\hskip
7.22743pt\mbox{ and }\\\
\mathbb{H}_{-k}^{c}:=\left\\{(r,c,\phi):-\frac{\pi}{2}+(k+1)^{-2}<\phi<-\frac{\pi}{2}+k^{-2}\right\\}\hskip
18.06749pt&\mbox{ with }\hskip
18.06749pt\mathbb{H}_{-k}:=\bigcup_{c\in\mathcal{C}_{0}}\mathbb{H}_{-k}^{c}.\end{split}$

The boundaries of the homogeneity strips are denoted by

$\mathbb{S}_{k}=\\{(r,c,\phi):|\phi|=\pi/2-k^{-2}\\}$

for $|k|\geq k_{0}$ with union

$\mathbb{S}=\bigcup_{|k|\geq k_{0}}\mathbb{S}_{k}.$

######  Definition 11.

A stable or unstable curve $W\subset\mathcal{D}$ is weakly homogeneous if $W$
is contained in one strip $\mathbb{H}_{k}$. An unstable manifold $W$ is
homogeneous if $f^{-n}(W)$ is weakly homogeneous for all $n\geq 0$. A stable
manifold $W$ is homogeneous if $f^{n}(W)$ is weakly homogeneous for all $n\geq
0$.

If $W$ is a weakly homogeneous stable or unstable curve contained in
$\mathbb{H}_{k}$ and contains the point $x=(r,c,\phi)$ then the length can be
bounded by (see [CM06, Lemma 5.10])

(23) $|W|\leq C(|k|+1)^{-3}\leq C\cos^{3/2}(\phi).$

As is standard in the theory of planar dispersive billiards, homogeneous
manifolds will allow us to keep control of distortion estimates. To this end,
following [CM06, 5.4], we introduced introduce a new collision space adapted
to the homogeneity strips.

For each $k\in\mathbb{Z}$ with $|k|\geq k_{0}$ and using (17), let

$\mathcal{D}_{i,k}:=\left(\partial
S_{i}\times\mathcal{C}_{i}\times[-\pi/2,\pi/2]\right)\cap\mathbb{H}_{k}.$

The new collision space $\mathcal{D}_{\mathbb{H}}$ is now defined to be the
disjoint union of the $\mathcal{D}_{i,k}$. As such, we have that
$\partial\mathcal{D}_{\mathbb{H}}=\mathbb{S}$. The map $f$ from the compact
model naturally acts on $\mathcal{D}_{\mathbb{H}}$ and this restriction is
denoted by $f_{\mathbb{H}}$, but note that it is not defined on
$f^{-1}(\mathcal{S}_{0})$ since $\mathcal{S}_{0}$ is not part of the new
collision space. In the new collision space we extend the set of
discontinuities (19) by defining for all $n\in\mathbb{N}$

$\Xi^{\mathbb{H}}_{n}:=\Xi_{n}\cup\left(\bigcup_{m=0}^{n}f^{-m}(\mathbb{S})\right),\hskip
9.03374pt\Xi^{\mathbb{H}}_{-n}:=\Xi_{-n}\cup\left(\bigcup_{m=0}^{n}f^{m}(\mathbb{S})\right)\hskip
9.03374pt\mbox{ and }\hskip
9.03374pt\Xi_{\pm\infty}^{\mathbb{H}}=\bigcup_{n\geq 0}\Xi_{\pm
n}^{\mathbb{H}}.$

Let $x\in\mathcal{D}\backslash\Xi_{\pm n}^{\mathbb{H}}$ and define
$\mathcal{Q}_{\pm n}^{t}(x)$ to be the path connected component of the set
$\mathcal{D}\backslash\Xi_{\pm n}^{\mathbb{H}}$ which contains $x$ (the
superscript $t$ is meant to denote tangential), and $\mathcal{Q}_{\mp n}(x)$
the union of all isometric connected components transversally close to
$\mathcal{Q}^{t}_{\mp n}(x)$. In other words, if
$x=(r,c,\phi)\in\mathcal{D}\backslash\Xi^{\mathbb{H}}_{\pm n}$, then there
exists a transverse neighborhood $U_{x}^{\pm
n}\subset\bigcup_{i}\mathcal{C}_{i}$ of $c$ such that

$\mathcal{Q}_{\pm n}(x)=\bigcup_{c^{\prime}\in U_{x}^{\pm
n}}\mathcal{Q}^{t}_{\pm n}(r,c^{\prime},\phi).$

Note that by compactness of $\mathcal{D}$ the set
$\mathcal{D}\backslash\Xi_{\pm n}^{\mathbb{H}}$ is the union of finitely many
sets of the form $\mathcal{Q}_{\pm n}(x)$. That is, for any $n\neq 0$ there
exist finitely many $p_{1},\dots,p_{k_{n}}$ such that

(24) $\mathcal{D}\backslash\Xi_{\pm
n}^{\mathbb{H}}=\bigcup_{i=1}^{k_{n}}\mathcal{Q}_{\pm n}(p_{i}).$

For a point $c$ in the ultrametric space $\bigcup_{i}\mathcal{C}_{i}$, let
$\mathcal{B}_{n}(c)$ be the ball of diameter $2^{-n}$ containing $c$.

For any $n\in\mathbb{N}$ we can further give a countable collection of sets

(25)
$\mathcal{Q}_{-n,m}(r,c,\phi):=\mathcal{Q}_{-n}(x)\cap(\partial\mathcal{S}_{0}\times\mathcal{B}_{m}(c)\times[-\pi/2,\pi/2]).$

Sets $\mathcal{Q}_{n,m}(x)$ are similarly defined for $n,m\in\mathbb{N}$. As
such, we have that

$\mathcal{Q}_{-n}^{t}(x)=\bigcap_{m>0}\mathcal{Q}_{-n,m}(x),\hskip
54.2025pt\mbox{ and }\hskip
54.2025pt\mathcal{Q}_{n}^{t}(x)=\bigcap_{m>0}\mathcal{Q}_{n,m}(x).$

Define for $x\in\mathcal{D}\backslash\Xi$ and $m\in\mathbb{N}$:

(26)
$\begin{split}\bar{W}^{cu}_{m}(x)&:=\bigcap_{n>0}\overline{\mathcal{Q}_{-n,m}(x)},\hskip
61.42993pt\hskip
68.65627pt\bar{W}^{u}(x):=\bigcap_{n>0}\overline{\bigcap_{m>0}\mathcal{Q}_{-n,m}(x)},\\\
\bar{W}^{cs}_{m}(x)&:=\bigcap_{n>0}\overline{\mathcal{Q}_{n,m}(x)},\hskip
61.42993pt\mbox{ and }\hskip
54.2025pt\bar{W}^{s}(x):=\bigcap_{n>0}\overline{\bigcap_{m>0}\mathcal{Q}_{n,m}(x)}\end{split}$

and let $W^{s}(x)$ and $W^{u}(x)$, respectively, be the curves
$\bar{W}^{s}(x)$ and $\bar{W}^{u}(x)$ without their endpoints. They are the
maximal homogeneous stable and unstable manifolds going through $x$.

######  Lemma 4.

For $x=(r,c,\phi)\in\mathcal{D}\backslash\Xi$, we have that

(27) $W^{u}(x)\subset\bigcap_{n>0}\mathcal{Q}^{t}_{-n}(r,c,\phi)\hskip
39.74872pt\mbox{ and }\hskip
39.74872ptW^{s}(x)\subset\bigcap_{n>0}\mathcal{Q}^{t}_{n}(r,c,\phi).$

Moreover if the boundary $\partial\mathcal{S}_{0}$ of the scatterers are
$C^{\ell}$ curves for some $\ell\geq 3$, then the curves $W^{u}(x)$ and
$W^{s}(x)$ are $C^{\ell-2}$ smooth with its derivatives uniformly bounded and
its $(\ell-2)^{nd}$ derivative Lipschitz with uniform constant.

###### Proof.

[CM06, §4.11]. ∎

Let $W$ be a smooth homogeneous unstable manifold and $p\in W$. The point $p$
divides $W$ into two subsegments, and denote by $r^{u}(p)=r_{W}(p)$ the length
(in the induced Euclidean metric from the flat metric on the leaves of the
tiling space $\Omega$) of the shorter one. If $W^{u}(x)=\varnothing$, then we
set $r(x)=0$. The proof for the periodic Lorenz gas with finite horizon
carries over with very minor modifications to the proof of the statement
[CM06, §5.5].

######  Theorem 5.

$W^{u}(x)$ exists (i.e. $r^{u}(x)>0$) for almost every $x\in\mathcal{D}$.
Moreover

(28) $\bar{\mu}(\\{x\in\mathcal{D}:r^{u}(x)<\varepsilon\\})<C\varepsilon$

for some constant $C$ independent of $x$ and for all $\varepsilon>0$.

###  4.1. Unstable densities and stable holonomies

The unstable partition $\\{\xi^{u}\\}$ of $\mathcal{D}$ consists of maximal
homogeneous unstable manifolds of points. More precisely, the element of the
partition containing a point $x$ is $\xi^{u}(x)=W^{u}(x)$ if
$W^{u}(x)\neq\varnothing$, and otherwise $\xi^{u}(x)=\\{x\\}$. Likewise the
stable partition $\\{\xi^{s}\\}$ is defined as $\xi^{s}(x)=W^{s}(x)$ if
$W^{s}(x)\neq\varnothing$, and otherwise $\xi^{s}(x)=\\{x\\}$. As in the
periodic case, the sets $\mathcal{Q}_{n,m}(x)$ in (25) form a countable
generator for the stable and unstable partitions. As such, the stable and
unstable partitions $\xi^{s},\xi^{u}$ are measurable.

Given that the stable and unstable partitions are measurable, it follows that
for $\mu$-almost every $x$ there are conditional measures induced by the
measure $\bar{\mu}$ on the stable and unstable manifolds $W=W^{s/u}(x)$
denoted respectively by $\nu_{W}$. As stated in Theorem 5.2 in [CM06], those
conditional measures are absolutely continuous with respect to the leaf volume
on $W$ with $C^{\ell-2}$ density $\rho_{W}$. We record here important
distortion estimates which follow from the analysis in [CM06, §5.6].

######  Proposition 3.

For any homogeneous unstable manifold $W$, for every $l=1,\dots,\ell-2$ there
is a $C_{l}>0$ such that

$\left|\frac{d^{l}}{dx^{l}}\ln\rho_{W}(x)\right|\leq\frac{C_{l}}{|W|^{2l/3}}.$

Moreover, for any $x,y\in W$,

$C_{1}^{-1}\leq e^{-C_{2}|W|^{1/3}}\leq\frac{\rho_{W}(x)}{\rho_{W}(y)}\leq
e^{C_{2}|W|^{1/3}}\leq C_{1}$

for some $C_{1},C_{2}>0$ which are independent of $W,x,y$.

######  Definition 12.

Let $W^{1},W^{2}$ be two unstable curves. Denote by

$W^{i}_{*}:=\\{x\in W^{i}:W^{s}(x)\cap W^{3-i}\neq\varnothing\\}$

for $i=1,2$. The map $\mathbf{h}:W^{1}_{*}\rightarrow W^{2}_{*}$ taking the
point $x\in W^{1}_{*}$ to $\mathbf{h}(x)=W^{s}(x)\cap W^{2}$ is called the
holonomy map.222In [CM06, Definition 5.41] the map $h$ is called the modified
holonomy map, since it is obtained by sliding along homogeneous stable
manifolds, hence it is a restriction of the usual holonomy map.

Note that for the holonomy map to be defined, it is necessary that the two
unstable curves lie on the same leaf of the foliated space $\Omega$. Suppose
the holonomy is defined at $x\in W^{1}$ with Lebesgue measure $m_{1}$ and
mapped to $W^{2}$ with Lebesgue measure $m_{2}$. Define the Jacobian of the
holonomy map to be

$J\mathbf{h}(x)=\frac{d\mathbf{h}^{-1}_{*}m_{2}}{dm_{1}}(x).$

######  Theorem 6.

There exists $C>1$ which depends only on the scatterers such that

$C^{-1}\leq j\mathbf{h}(x)\leq C$

for all $x\in W^{1}_{*}$. In addition, there is an $A>1$ which also only
depends on the model scatterers such that

$A^{-\gamma-\delta^{1/3}}\leq J\mathbf{h}(x)\leq A^{\gamma+\delta^{1/3}},$

where $\delta$ is the distance between $x$ and $\mathbf{h}(x)$ and $\gamma$
the angle between the tangent vectors to the curves $W^{1}$ and $W^{2}$ at the
points $x$ and $\mathbf{h}(x)$, respectively.

The proof follows the proof of the periodic case, [CM06, Theorem 5.42].

The next result establishes that the holonomy map is leafwise dynamically
Hölder continuous, that is Hölder continuous with respect to the separation
time defined on the leaves of foliation of $\mathcal{D}$ given by the orbits
of the $\mathbb{R}^{2}$-action. For two points $x,y\in\mathcal{D}$ on the same
leaf define

$s_{+}(x,y):=\min\\{n\geq 0~{}|~{}y\not\in\mathcal{Q}^{t}_{n}(x)\\}.$

The following can be proven using identical arguments as Proposition 5.48 in
[CM06].

######  Proposition 4.

There are constants $C>0$ and $\theta\in(0,1)$ such that

$|\ln J\mathbf{h}(x)-\ln J\mathbf{h}(y)|\leq C\theta^{s_{+}(x,y)}.$

Let $W$ be a weakly homogeneous curve and $m_{W}$ the Lebesgue measure on it.
For any $x\in W$ and $n\geq 0$ let $W_{n}(x)$ be the H-component of $f^{n}(W)$
containing $f^{n}(x)$ and by $r_{n}(x)=r_{W_{n}(x)}(f^{n}(x))$ the distance
from $f^{n}(x)$ to the nearest point of $\partial W_{n}(x)$. We now state the
two growth lemmas, the proofs of which follow from the proof for the periodic
case [CM06, §5.10].

######  Theorem 7.

There exist constants $\hat{\Lambda}>1$, $\theta_{1}\in(0,1)$, $c_{1},c_{2}>0$
such that for any $n\geq 0$ and $\varepsilon>0$,

(29) $m_{W}(r_{n}(x)<\varepsilon)\leq
c_{1}(\theta_{1}\hat{\Lambda})^{n}m_{W}(r_{0}(x)<\varepsilon/\hat{\Lambda}^{n})+c_{2}\varepsilon
m_{W}(W).$

In addition there are constants $\zeta>0$ and $c_{3}>0$ such that for all
$n\geq\zeta\left|\ln|W|\right|$ and $\varepsilon>0$,

(30) $m_{W}(r_{n}(x)<\varepsilon)\leq c_{3}\varepsilon m_{W}(W).$

Finally, we state the analog of Sinai’s fundamental theorem for dispersing
billiards [CM06, Theorem 5.70].

######  Theorem 8.

Let $x\in\mathcal{D}\backslash\Xi_{\infty}^{\mathbb{H}}$. For any $q>0$ and
$A>0$ there exists an leafwise open neighborhood
$\mathcal{U}^{u}_{x}\subset\mathcal{D}$ of $x$ such that for any unstable
curve $W\subset\mathcal{U}_{x}^{u}$,

$m_{W}(y\in W:r^{s}(y)>A|W|)\geq(1-q)m_{W}(W).$

Additionally, for $x\in\mathcal{D}\backslash\Xi_{-\infty}^{\mathbb{H}}$, any
$q>0$ and $A>0$ there exists a leafwise open neighborhood
$\mathcal{U}^{s}_{x}\subset\mathcal{D}$ of $x$ such that for any stable curve
$W\subset\mathcal{U}_{x}^{s}$,

$m_{W}(y\in W:r^{u}(y)>A|W|)\geq(1-q)m_{W}(W).$

##  5\. Ergodic properties

The goal of this section is to prove the following.

######  Theorem 9.

Let $\mathcal{T}$ be a repetitive, aperiodic tiling of $\mathbb{R}^{2}$ of
finite local complexity, $\mathcal{S}$ be a $\mathcal{T}$-equivariant
collection of dispersive scatterers with $C^{3}$ boundary defining a Lorentz
gas of category A, and $\mu$ an $\mathbb{R}^{2}$-invariant ergodic probability
measure on $\Omega_{\mathcal{T}}$. Then $(f,\mathcal{D},\bar{\mu})$ has the K
property.

Let us summarize the strategy of the proof in this section. Our goal will be
to show that the Pinsker partition is trivial, which is one characterization
of a K-system (recall that the Pinsker partition is the join of all partitions
of zero entropy). In order to do this, we will first show that $\mu$-almost
every connected component is fully contained (up to a subset of Lebesgue
measure zero) in a single element of the Pinsker partition. Using the
(minimal) foliated structure of the tiling space and the dynamics of $f$, it
will follow that a dense union of connected components is contained in an
element of the Pinsker partition. Finally, using again the structure of the
tiling space and the ergodicity of $\mu$ we will show that the Pinsker
partition is trivial. We denote the Pinsker partition of $f$ by $\Pi_{f}$ and
by $\Pi_{f}(x)$ the element of the Pinsker partition containing $x$.

######  Proposition 5.

$\mu$-almost every connected component $\mathcal{L}\subset\mathcal{D}$ is
contained (mod 0) with respect to the Lebesgue measure in one element of the
Pinsker partition.

Before we proceed with the proof of Proposition 5, let us recall the partition
$\xi^{u}$ of $\mathcal{D}$ into homogeneous unstable manifolds. This partition
satisfies [CM06, §5.1]:

(31) $f^{-1}\xi^{u}\geq\xi^{u},\hskip
54.2025pt\bigvee_{i\in\mathbb{Z}}f^{i}\xi^{u}=\varepsilon,\hskip
54.2025pt\mbox{ and consequently,}\hskip
54.2025pt\bigwedge_{i\in\mathbb{Z}}f^{i}\xi^{u}\geq\Pi_{f}.$

Replacing the map $f^{-1}$ with $f$, an analogous statement holds for the
stable partition $\xi^{s}$.

The following is an important consequence of Theorem 8; the proof follows from
the periodic case [CM06, §6.5].

######  Theorem 10.

For any point
$x\in\mathcal{D}\backslash(\Xi_{\infty}^{\mathbb{H}}\cup\Xi_{-\infty}^{\mathbb{H}})$
there is a leafwise open neighborhood $\mathcal{U}_{x}\subset\mathcal{D}$ with
the property that Lebesgue almost every pair of points in $\mathcal{U}_{x}$
can be connected by a sequence of homogenous stable and unstable manifolds
contained in $\mathcal{U}_{x}$. Moreover, if $x\in\mathcal{D}$ lies on exactly
one smooth singularity curve
$S\subset\Xi_{\infty}^{\mathbb{H}}\cup\Xi_{-\infty}^{\mathbb{H}}$ then there
is a leafwise open neighborhood $\mathcal{U}_{x}\subset\mathcal{D}$ with the
same property.

We also need the following observation, the proof of the which is the same as
in the periodic case [CM06, Lemma 6.18].

######  Lemma 5.

The set of points which belong on more than one smooth curve of the
singularity set $\Xi_{\infty}^{\mathbb{H}}\cup\Xi_{-\infty}^{\mathbb{H}}$ is
leafwise countable, that is, for every connected component
$\mathcal{L}\subset\mathcal{D}$, the subset
$\mathcal{B}_{\mathcal{L}}\subset\mathcal{L}$ consisting of points which
belong to more than one singularity curve is countable.

###### Proof of Proposition 5.

It follows from Lemma 5 that for any $x\in\mathcal{D}$ and two points
$y,z\in\mathcal{L}_{x}\setminus\mathcal{B}_{\mathcal{L}_{x}}$ there exists a
path
$\gamma_{y,z}\subset\mathcal{L}_{x}\setminus\mathcal{B}_{\mathcal{L}_{x}}$
from $y$ to $z$. Picking $w\in\gamma_{y,z}$ and appealing to Theorem 10 there
is a leafwise open neighborhood $\mathcal{U}_{w}$ such that Leb-a.e. two
points in $\mathcal{U}_{w}$ can be joined by a sequence of stable and unstable
manifolds. Noting that $\gamma_{y,z}$ is a compact set, we can pick finitely
many points $w_{1},\dots,w_{N}$ such that the union of their corresponding
neighborhoods $\mathcal{U}_{w_{1}},\dots,\mathcal{U}_{w_{N}}$ covers
$\gamma_{y,z}$. By the last property in (31) and the local product structure
of $\bar{\mu}$, it follows that for $\bar{\mu}$-almost every $x$ and Leb-a.e.
$\omega\in\mathcal{L}_{x}$,

(32) $\xi^{u}(\omega)\subset\Pi_{f}(\omega)\hskip 72.26999pt\mbox{ and }\hskip
72.26999pt\xi^{s}(\omega)\subset\Pi_{f}(\omega).\;\;$

Therefore by (32) it follows that $y,z$, $\gamma_{y,z}$ and in fact the entire
neighborhood of $\gamma_{y,z}$ defined by the union of the neighborhoods
$\mathcal{U}_{w_{1}},\dots,\mathcal{U}_{w_{N}}$ is contained up to Lebesgue
measure zero in the same element of the Pinsker partition
$\Pi_{f}(y)=\Pi_{f}(z)$. Since we can apply the same argument to almost any
two points on a connected component, we deduce that the entire connected
component is contained up to Lebesgue measure zero in one element of the
Pinsker partition. ∎

######  Proposition 6.

For almost every connected component $\mathcal{L}$,
$\Pi_{f}(\mathcal{L})=\Pi_{f}(f(\mathcal{L}))$.

We will need the following Lemma, the proof of which is contained in the proof
of [CM06, Lemma 6.23].

######  Lemma 6.

For any connected component $\mathcal{L}\subset\mathcal{D}$ there are two
other connected components $\mathcal{L}_{1}$,
$\mathcal{L}_{2}\subset\mathcal{D}$ such that the intersections

$f(\mathcal{L})\cap\mathcal{L}_{1},\hskip
25.29494ptf(\mathcal{L})\cap\mathcal{L}_{2},\hskip
25.29494ptf(\mathcal{L}_{1})\cap\mathcal{L}_{2},\hskip 25.29494pt\mbox{ and
}\hskip 25.29494ptf(\mathcal{L}_{1})\cap\mathcal{L}$

are all non-empty and have positive Lebesgue measure.

###### Proof of Proposition 6.

We first recall that the Pinsker partition satisfies $\Pi_{f}\geq\xi$ for any
measurable partition $\xi$ with zero entropy. Using this property and the fact
that $f(\Pi_{f}),f^{-1}(\Pi_{f})$ have zero entropy it can be deduced that the
Pinsker partition satisfies the invariance condition
$\Pi_{f}(f(x))=f(\Pi_{f}(x))$ for almost every $x$. Denote by
$\Pi_{f}(\mathcal{L})$ the element of the Pinsker partition which, by
Proposition 5, contains the connected component $\mathcal{L}$. We then have
that $f(\Pi_{f}(\mathcal{L}))=\Pi_{f}(f(\mathcal{L}))$.

It will be usefull to remember that a connected component
$\mathcal{L}\subset\mathcal{D}$ corresponds to a specific scatterer $\Upsilon$
in a specific tiling $\mathcal{T}\in\Omega$ along with all of its possible
incoming/outgoing collisions. Let $\mathcal{L}_{1},\mathcal{L}_{2}$ be as in
Lemma 6. From the point of view of the Lorentz gas this corresponds to
scatterers $\Upsilon_{1},\Upsilon_{2}$ on $\mathcal{T}$ along with all
possible incoming/outgoing collisions such that $\Upsilon$ sends an open set
of collisions to $\Upsilon_{1}$ and $\Upsilon_{2}$, and in turn $\Upsilon_{1}$
sends an open set of collisions to $\Upsilon$ and $\Upsilon_{2}$.

The first part implies that there are subsets $A_{i}\subset\mathcal{L}_{i}$ of
positive Lebesgue measure, $i=1,2$, such that $A_{1}\cup A_{2}\subset
f(\Pi_{f}(\mathcal{L}))=\Pi_{f}(f(\mathcal{L}))$. However, Proposition 5
forces the entire connected component to be included, and so
$\mathcal{L}_{1}\cup\mathcal{L}_{2}\subset
f(\Pi_{f}(\mathcal{L}))=\Pi_{f}(f(\mathcal{L}))$. In other words,
$\mathcal{L}_{1}\cup\mathcal{L}_{2}$ is contained in one element of the
Pinsker partition.

From the second part it follows that $\mathcal{L}\subset
f(\Pi_{f}(\mathcal{L}_{1}))=\Pi_{f}(f(\mathcal{L}_{1}))$ and
$\mathcal{L}_{2}\subset
f(\Pi_{f}(\mathcal{L}_{1}))=\Pi_{f}(f(\mathcal{L}_{1}))$, so that
$\mathcal{L}\cup\mathcal{L}_{2}$ is contained in one element of the Pinsker
partition. Thus we have that
$\Pi_{f}(f(\mathcal{L}))=\Pi_{f}(\mathcal{L}_{1})=\Pi_{f}(\mathcal{L}_{2})=\Pi_{f}(\mathcal{L})$.
∎

For $x\in\mathcal{D}$, let $\mathbb{L}_{x}\subset\mathcal{D}$ be the union of
connected components intersecting the $\mathbb{R}^{2}$-orbit of
$\pi(x)\in\Omega$. In other words, this is the union of all scatterers for the
Lorenz gas on $\mathcal{T}=\pi(x)$, along with all possible incoming and
outgoing collisions. Proposition 5 and Proposition 6 now have the following
consequence.

######  Corollary 3.

For $\bar{\mu}$-almost every $x\in\mathcal{D}$ the element of the Pinsker
partition containing $x$ contains all the scatterers for the aperiodic Lorenz
gas defined on $\pi(x)\in\Omega$. That is,

$\Pi_{f}(\mathbb{L}_{x})\subset\Pi_{f}(x).$

###### Proof of Theorem 9.

Corollary 3 implies that any element of the Pinsker partition is the union of
Poincaré sections of Lorenz gases of different tilings in $\Omega$. In other
words, for $\bar{\mu}$-almost every $x\in\mathcal{D}$ there is a subset
$\mathcal{K}_{x}\subset\bigcup_{i}\mathcal{C}_{i}$ such that:

  1. (i)

$\mathcal{K}_{x}$ is the intersection of the canonical transversal $\mho$ with
an $\mathbb{R}^{2}$-invariant set:

$\mathcal{K}_{x}:=\mho~{}\cap~{}\bigcup_{p\in\Pi_{f}(x)}\bigcup_{v\in\mathbb{R}^{2}}\varphi_{v}(\pi(p)),$

  2. (ii)

$\Pi_{f}(x)=\bigcup_{p\in\mathcal{K}_{x}}\mathbb{L}_{p},$ and

  3. (iii)

$\bar{\mu}(\Pi_{f}(x))=2\sum_{i}\nu(\mathcal{K}_{x}\cap\mathcal{C}_{i})\cdot\mathrm{Leb}(\partial\mathcal{S}^{i}),$

where we have used the local product structure of $\bar{\mu}$. We now recall
that the measure $\bar{\mu}$ is obtained from an $\mathbb{R}^{2}$-invariant
ergodic probability measure $\mu$ on $\Omega$ with local product structure
$\mathrm{Leb}\times\nu$. Ergodicity of $\mu$ implies that the set
$\mathcal{K}_{x}$ either has full or null $\nu$-measure. Consequently,
$\bar{\mu}(\Pi_{f}(x))\in\\{0,\bar{\mu}(\mathcal{D})\\}$, which implies that
the Pinsker partition is trivial. ∎

##  6\. Planar mixing for aperiodic Lorentz gases

Throughout this section we assume that $\mu$ is an $\mathbb{R}^{2}$-invariant
ergodic probability measure on a tiling space $\Omega$ of an aperiodic,
repetitive tiling of $\mathbb{R}^{2}$ of finite local complexity and that
$\mathcal{T}\in\Omega$ is a generic tiling with respect to this measure. In
particular all results in this section apply to any aperiodic, repetitive
tiling of $\mathbb{R}^{2}$ of finite local complexity and uniform patch
frequency as such tilings result in uniquely ergodic $\mathbb{R}^{2}$ action
on $\Omega_{\mathcal{T}}$.

Recall that $\mu=\mbox{Vol}\times\nu$ and that by Proposition 2, the
corresponding $f$-invariant measure on the collision space $\mathcal{D}$ is
given by $\bar{\mu}=\cos\theta\,d\rho\,d\nu\,d\theta$.

######  Definition 13.

If $\mathcal{S}\subset\mathbb{R}^{2}$ is a collection of
$\mathcal{T}$-equivariant scatterers, then a function
$g^{\prime}:\partial\mathcal{S}\times X\rightarrow\mathbb{R}$ is
$\mathcal{T}$-equivariant if there exists a $\mathcal{T}$-equivariant function
$g:\mathbb{R}^{2}\times X\rightarrow\mathbb{R}$ such that
$g|_{\partial\mathcal{S}}=g^{\prime}$, that is, $g^{\prime}$ is the
restriction to $\partial\mathcal{S}$ of $g$. A $\mathcal{T}$-equivariant
function $g:\partial\mathcal{S}\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ is
locally integrable if for any connected component
$\mathcal{L}\subset\partial\mathcal{S}$ we have that
$\int_{-\pi/2}^{\pi/2}\int_{\mathcal{L}}g\,d\rho\cos\theta\,d\theta$ exists.

We will denote the set of locally integrable, $\mathcal{T}$-equivariant
functions on $\partial\mathcal{S}\times(-\pi/2,\pi/2)$ by
$\Delta_{\mathcal{T}}(\mathcal{S})$.

######  Lemma 7.

For $\mu$\- almost every $\mathcal{T}\in\Omega$ and any
$g\in\Delta_{\mathcal{T}}(\mathcal{S})$ and $z\in\mathbb{R}^{2}$

(33)
$\beta(g):=\lim_{T\rightarrow\infty}\frac{1}{\mbox{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(z)\cap\partial\mathcal{S}}g\,d\rho\,\cos\theta\,d\theta$

exists and is independent of $z\in\partial\mathcal{S}$. In addition, there
exists a transversally locally constant function
$h:\partial\Sigma\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ such that
$g(v,\theta)=h(\varphi_{v}(\mathcal{T}),\theta)$ for all
$\theta\in(-\pi/2,\pi/2)$ and $v\in\mathbb{R}^{2}$ such that
$\varphi_{v}(\mathcal{T})\in\partial\mathcal{S}$, and

$\beta(g)=\int_{-\pi/2}^{\pi/2}\int_{\partial\Sigma}h\,d\rho\,d\nu\,\cos\theta\,d\theta=\bar{\mu}(h).$

The quantity $\beta(g)$ in (33) is called the planar average of $g$.

###### Proof.

Let $\Gamma\subset\mathbb{R}^{2}$ be a $C^{3}$ closed convex curve whose
barycenter coincides with the origin. Then there exists an
$\varepsilon_{\Gamma}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{\Gamma})$ the $\varepsilon$-neighborhood of
$\Gamma$, the open annulus $\mathcal{A}_{\Gamma}^{\varepsilon}$, can be given
coordinates $(\rho,y)$ for $\rho\in S^{1}$ and
$y\in(-\varepsilon,\varepsilon)$. As such, there exists a function
$J:\mathcal{A}_{\Gamma}^{\varepsilon}\rightarrow\mathbb{R}$, positive in the
interior, depending on the curvature of $\Gamma$, such that for any measurable
function $f$ with support in $\mathcal{A}_{\Gamma}^{\varepsilon}$ we have

$\int_{\mathcal{A}_{\Gamma}^{\varepsilon}}f\,d\mbox{Vol}=\int_{\Gamma}\int_{-\varepsilon}^{\varepsilon}f(\rho,y)J(\rho,y)\,dy\,d\rho.$

Let $\omega:(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}$ be a non-negative
smooth bump function with $\omega(0)=1$ and integral 1, and consider the
function $u:S^{1}\times(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}$
defined as $u(\rho,y)=\frac{\omega(y)}{J(\rho,y)}$. Now for any
$g:\Gamma\rightarrow\mathbb{R}$, let
$g_{\omega}:\mathcal{A}_{\Gamma}^{\varepsilon}\rightarrow\mathbb{R}$ be
defined as $g_{\omega}(\rho,y):=g(\rho)u(\rho,y)$. As such,

(34)
$\begin{split}\int_{\mathcal{A}_{\Gamma}^{\varepsilon}}g_{\omega}\,d\mbox{Vol}&=\int_{\Gamma}\int_{-\varepsilon}^{\varepsilon}g_{\omega}(\rho,y)J(\rho,y)\,dy\,d\rho=\int_{\Gamma}\int_{-\varepsilon}^{\varepsilon}g(\rho)u(\rho,y)J(\rho,y)\,dy\,d\rho\\\
&=\int_{\Gamma}\int_{-\varepsilon}^{\varepsilon}g(\rho)\omega(y)\,dy\,d\rho=\int_{\Gamma}g(\rho)\,d\rho.\end{split}$

Given a $\mathcal{T}$-equivariant collection of $C^{3}$ scatterers
$\mathcal{S}$, there is a finite set of geometrically different closed convex
curves such that any connected component $\Gamma\subset\partial\mathcal{S}$ is
translation-equivalent to one of these curves. Thus we can find an
$\varepsilon_{\mathcal{S}}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{\mathcal{S}})$, the $\varepsilon$-neighborhood
of any connected component of $\partial\mathcal{S}$ admits coordinates in
$S^{1}\times(-\varepsilon,\varepsilon)$, and thus the
$\varepsilon$-neighborhood of $\partial\mathcal{S}$ admits coordinates in
$\mathcal{S}\times(-\varepsilon,\varepsilon)$. Again by finite local
complexity, there is a finite set of functions
$\mathcal{J}_{\varepsilon}(\mathcal{S})=\\{J_{i}\\}$, one for each translation
class of scatterers in $\mathcal{S}$, and a bump function
$\omega_{\mathcal{S}}:(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}$ so that
using the $J_{i}$ and $\omega_{\mathcal{S}}$ as above we can extend a function
$g\in\Delta_{\mathcal{T}}(\mathcal{S})$ to a $\mathcal{T}$-equivariant
function $g_{\omega_{\mathcal{S}}}$ defined on the $\varepsilon$-neighborhood
of $\partial\mathcal{S}$ in a way that preserves the integral as in (34). More
specifically, we have that

(35)
$\int_{-\pi/2}^{\pi/2}\int_{B_{T}(z)\cap\partial\mathcal{S}}g\,d\rho\,\cos\theta\,d\theta=\int_{-\pi/2}^{\pi/2}\int_{B_{T}(z)}g_{\omega_{\mathcal{S}}}\,d\mbox{Vol}\,\cos\theta\,d\theta+\mathcal{O}(|\partial
B_{T}(z)|),$

where the error term comes from $\varepsilon$-neighborhoods of connected
components of $\partial\mathcal{S}$ that intersect $\partial B_{T}$.

Since $g_{\omega_{\mathcal{S}}}$ is $\mathcal{T}$-equivariant on
$\mathbb{R}^{2}\times(-\pi/2,\pi/2)$, by Corollary 2, there exists a
transversally locally constant function
$h_{\omega}:\Omega\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ such that
$g_{\omega_{\mathcal{S}}}(v,\theta)=h_{\omega}(\varphi_{v}(\mathcal{T}),\theta)$
for all $v\in\mathbb{R}^{d}$ and $\theta\in(-\pi/2,\pi/2)$, and we have that

(36)
$\begin{split}\lim_{T\to\infty}\frac{1}{\mbox{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(z)\cap\partial\mathcal{S}}g\,d\rho\,\cos\theta\,d\theta&=\lim_{T\to\infty}\frac{1}{\mbox{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(z)}g_{\omega_{\mathcal{S}}}\,d\mbox{Vol}\,\cos\theta\,d\theta\\\
&=\int_{-\pi/2}^{\pi/2}\int_{\Omega}h_{\omega}(\mathcal{T}^{\prime},\theta)d\mu(\mathcal{T^{\prime}})\cos\theta\,d\theta\end{split}$

proving the first part of the lemma. To show the second part of the lemma we
observe that by the definition of $g_{\omega_{\mathcal{S}}}$, we have that
$h_{\omega}$ is supported in the $\varepsilon$-neighborhood of
$\partial\Sigma\times(-\pi/2,\pi/2)$, and this set admits coordinates in
$\partial\Sigma\times(-\pi/2,\pi/2)\times(-\varepsilon,\varepsilon)$.
Furthermore, without loss of generality we refine the coordinates for
$\partial\Sigma$ given in (17) as

$\partial\Sigma=\bigsqcup_{i}\Gamma_{i}\times\mathcal{C}_{i}^{\prime}$

in such a way that

  1. (i)

$\Gamma_{i}$ is translation equivalent to a connected component of
$\mathcal{S}$,

  2. (ii)

$h$ is transversally constant in each $\mathcal{C}_{i}^{\prime}$, and

  3. (iii)

$J_{i}^{\prime}\in\mathcal{J}_{\varepsilon}(\mathcal{S})$ is the corresponding
function with which to integrate around $\Gamma_{i}$ as in (34).

Using this and letting $\Gamma_{i}^{\varepsilon}$ denote the
$\varepsilon$-neighborhood of $\Gamma_{i}$, we have as in (34),

$\begin{split}\int_{-\pi/2}^{\pi/2}\int_{\Omega}h_{\omega}(\mathcal{T},\theta)\,d\mu(\mathcal{T})\,\cos\theta\,d\theta&=\int_{-\pi/2}^{\pi/2}\sum_{i}\int_{\mathcal{C}_{i}^{\prime}}\int_{\Gamma_{i}^{\varepsilon}}h_{\omega}\,d\mathrm{Leb}\,d\nu\,\cos\theta\,d\theta\\\
&=\int_{-\pi/2}^{\pi/2}\sum_{i}\int_{\mathcal{C}_{i}^{\prime}}\int_{\Gamma_{i}}\int_{-\varepsilon}^{\varepsilon}h(\rho)\frac{\omega_{\mathcal{S}}(y)}{J_{i}^{\prime}(\rho,y)}\,dy\,d\rho\,d\nu\,\cos\theta\,d\theta\\\
&=\int_{-\pi/2}^{\pi/2}\sum_{i}\int_{\mathcal{C}_{i}^{\prime}}\int_{\Gamma_{i}}h(\rho)\,d\rho\,d\nu\,\cos\theta\,d\theta\\\
&=\int_{-\pi/2}^{\pi/2}\int_{\partial\Sigma}h\,d\rho\,d\nu\,\cos\theta\,d\theta=\bar{\mu}(h)\end{split}$

which in conjunction with (36) gives the result. ∎

For $\theta\in(-\pi/2,\pi/2)$ and $n\in\mathbb{N}$ we denote by
$F^{n}_{\theta}:\partial\mathcal{S}\rightarrow\partial\mathcal{S}$ the map
given by the first $n$ colisions of the planar Lorentz gas with initial angle
$\theta\in(-\pi/2,\pi/2)$. The map $F^{n}_{\theta}$ is not defined on all of
$\partial\mathcal{S}\times(-\pi/2,\pi/2)$ due to discontinuities, but this is
a set of Lebesgue measure zero for all $n\in\mathbb{N}$.

######  Theorem 11.

Let $\Omega$ be the tiling space of an aperiodic, repetitive tiling
$\mathcal{T}^{\prime}$ of $\mathbb{R}^{2}$ of finite local complexity and let
$\mu$ be an invariant ergodic probability measure for the $\mathbb{R}^{2}$
action. Let $\partial\Sigma\subset\Omega$ be a transversally locally constant
set which defines for every $\mathcal{T}\in\Omega$ a $\mathcal{T}$-equivariant
collection $\mathcal{S}_{\mathcal{T}}\subset\mathbb{R}^{2}$ of dispersive
scatterers with $C^{3}$-boundary of category A.

For $\mu$-almost every $\mathcal{T}\in\Omega$, for
$g_{1},g_{2}\in\Delta_{\mathcal{T}}(\mathcal{S}_{\mathcal{T}})$ there exist
transversally locally constant functions
$h_{1},h_{2}:\mathcal{D}\rightarrow\mathbb{R}$ such that for any
$n\in\mathbb{N}$ and all $T$ large enough (depending on $n$), we have that for
any $x\in\mathbb{R}^{2}$

(37)
$\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F^{n}_{\theta}\cdot
g_{2}\,d\rho\,\cos\theta\,d\theta=\mathrm{Vol}(B_{T})\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}+o_{n}(T^{2}),$

from which we obtain the convergence

(38)
$\lim_{n\rightarrow\infty}\lim_{T\rightarrow\infty}\frac{1}{\mathrm{Vol}(B_{T})}\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F_{\theta}^{n}\cdot
g_{2}\,d\rho\,\cos\theta\,d\theta=\beta(g_{1})\beta(g_{2}).$

In the case of substitution tilings, we have the following.

######  Theorem 12.

Let $\Omega$ be the tiling space of a self-similar tiling
$\mathcal{T}^{\prime}$ of $\mathbb{R}^{2}$ of finite local complexity and let
$\mu$ be the unique measure invariant under the $\mathbb{R}^{2}$ action. Let
$\partial\Sigma\subset\Omega$ be a transversally locally constant set which
defines for every $\mathcal{T}\in\Omega$ a $\mathcal{T}$-equivariant
collection $\mathcal{S}_{\mathcal{T}}\subset\mathbb{R}^{2}$ of dispersive
scatterers with $C^{3}$-boundary of category A.

For any $n$ there exist $d^{+}\geq 1$ functions
$C_{i}^{n}:\Delta_{\mathcal{T}}(\mathcal{S})\times\Delta_{\mathcal{T}}(\mathcal{S})\rightarrow\mathbb{R}$
(where $d^{+}$ depends on the spectrum of the substitution matrix) such that
for any $\mathcal{T}\in\Omega$, for
$g_{1},g_{2}\in\Delta_{\mathcal{T}}(\mathcal{S}_{\mathcal{T}})$ there exist
transversally locally constant functions
$h_{1},h_{2}:\mathcal{D}\rightarrow\mathbb{R}$ such that for any
$n\in\mathbb{N}$ and all $T$ large enough (depending on $n$), we have that

(39)
$\begin{split}&\int_{-\pi/2}^{\pi/2}\int_{B_{T}\cap\partial\mathcal{S}_{\mathcal{T}}}g_{1}\circ
F^{n}_{\theta}\cdot g_{2}\,d\rho\,\cos\theta\,d\theta\\\ &\hskip
36.135pt=\sum_{i=1}^{d^{+}}L_{i}(T)T^{2\frac{\log|\lambda_{i}|}{\log\lambda_{1}}}C_{i}^{n}(g_{1},g_{2})+\mathcal{O}_{n}(T),\\\
&\hskip 36.135pt=\mathrm{Vol}(B_{T})\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}+\sum_{i=2}^{d^{+}}L_{i}(T)T^{2\frac{\log|\lambda_{i}|}{\log\lambda_{1}}}C_{i}^{n}(g_{1},g_{2})+\mathcal{O}_{n}(T),\end{split}$

where $L_{i}(T)=\log(T)^{s-1}$ is the size of the Jordan block associated to
the $i^{th}$ eigenvalue of the substitution matrix is $s$. Moreover, the
planar averaged mixing (38) also holds.

###### Proof of Theorem 11.

Let $\mathcal{T}\in\Omega$ be a $\mu$-generic tiling and $\mathcal{S}$ a
$\mathcal{T}$-equivariant collection of dispersive scatterers with
$C^{3}$-boundary of category A with $\bar{0}\in\partial\mathcal{S}$. If
$g_{1},g_{2}\in\Delta_{\mathcal{T}}(\mathcal{S})$, then by Lemma 7, there
exist transversally locally constant functions
$h_{1},h_{2}:\mathcal{D}\rightarrow\mathbb{R}$ such that:
$g_{i}(v)=h_{i}\circ\varphi_{v}(\mathcal{T})$ for all
$\theta\in(-\pi/2,\pi/2)$ and $v\in\mathbb{R}^{2}$ such that
$\varphi_{v}(\mathcal{T})\in\partial\mathcal{S}$, and
$\beta(g_{i})=\bar{\mu}(h_{i})$. For any $n\in\mathbb{N}$, the function
$g_{1}\circ F_{\theta}^{n}\cdot g_{2}$ is a locally integrable
$\mathcal{T}$-equivariant function. In fact, it is the product of two locally
integrable $\mathcal{T}$-equivariant functions $g_{1}\circ F_{\theta}^{n}$ and
$g_{2}$, and, by construction, the associated transversally locally functions
on $\mathcal{D}$ are $h_{1}\circ f^{n}$ and $h_{2}$.

By the second part of Lemma 7, we have that

(40) $\begin{split}\beta(g_{1}\circ F_{\theta}^{n}\cdot
g_{2})&=\int_{-\pi/2}^{\pi/2}\int_{\partial\Sigma}h_{1}\circ f^{n}\cdot
h_{2}\,d\rho\,d\nu\,\cos\theta\,d\theta\\\ &=\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})},\end{split}$

Thus, for large $T$ we have that

(41)
$\begin{split}&\int_{-\pi/2}^{\pi/2}\int_{B_{T}(x)\cap\partial\mathcal{S}}g_{1}\circ
F_{\theta}^{n}\cdot g_{2}\,d\rho\,\cos\theta\,d\theta\\\ &\hskip
108.405pt=\mathrm{Vol}(B_{T})\int_{-\pi/2}^{\pi/2}\int_{\partial\Sigma}h_{1}\circ
f^{n}\cdot h_{2}\,d\rho\,d\nu\cos\theta\,d\theta+o_{n}(T^{2})\\\ &\hskip
108.405pt=\mathrm{Vol}(B_{T})\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}+o_{n}(T^{2}),\end{split}$

proving (37). The leafwise mixing result (38) follows from (40) by applying
Theorem 9. Indeed we have

$\lim_{n\to\infty}\beta(g_{1}\circ F_{\theta}^{n}\cdot
g_{2})=\lim_{n\to\infty}\langle h_{1}\circ
f^{n},h_{2}\rangle_{L^{2}_{\bar{\mu}}(\mathcal{D})}=\bar{\mu}(h_{1})\bar{\mu}(h_{2})=\beta(g_{1})\beta(g_{2}).$

∎

###### Proof of Theorem 12.

To go from (37) to (39) we need to establish rates of convergence for ergodic
integrals of locally integrable, $\mathcal{T}$-equivariant functions. This
essentially is done in [BS13], giving error rates for ergodic integrals for
locally integrable, $\mathcal{T}$-equivariant functions, giving (39) the same
way that (41) was obtained. ∎

##  7\. Ergodicity of the flow

Theorem 9, which in particular implies the ergodicity of the billiard map,
also implies the ergodicity of the billiard flow. In this section we prove
Theorem 4.

Let $\mathcal{T}$ be a repetitive, aperiodic tiling of $\mathbb{R}^{2}$ of
finite local complexity, $\mathcal{S}$ be a $\mathcal{T}$-equivariant
collection of dispersive scatterers with $C^{3}$ boundary of category A, and
$\mu$ an $\mathbb{R}^{2}$-invariant ergodic probability measure on
$\Omega_{\mathcal{T}}$. Let
$h:\Omega\times(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ be a $L^{1}$ transversally
locally constant function which defines for every
$\mathcal{T}^{\prime}\in\Omega$ a
$g_{\mathcal{T}^{\prime}}\in\mathfrak{G}_{\mathcal{T}^{\prime}}$ by
$g_{\mathcal{T}^{\prime}}(z,\theta)=h(\varphi_{z}(\mathcal{T}^{\prime}),\theta)$,
where $\mathfrak{G}_{\mathcal{T}}$ is defined in (6). By the relation between
$\bar{\phi}_{t}(\mathcal{T}^{\prime},\theta)$ and
$\phi_{t}^{\mathcal{T^{\prime}}}(\bar{0},\theta)$ described in Section 3.2, we
have for all $t\in\mathbb{R}$ that

(42)
$h\circ\bar{\phi}_{t}(\mathcal{T}^{\prime},\theta)=g_{\mathcal{T}^{\prime}}\circ\phi_{t}^{\mathcal{T^{\prime}}}(\bar{0},\theta).$

Consequently, defining a transversally locally constant function
$\bar{H}:\mathcal{D}\to\mathbb{R}$ by

$\bar{H}(\mathcal{T}^{\prime},\theta):=\int_{0}^{\tau(\mathcal{T}^{\prime},\theta)}h\circ\bar{\phi}_{t}(\mathcal{T}^{\prime},\theta)dt,$

we obtain, recalling (8), that

$\bar{H}(\mathcal{T}^{\prime},\theta)=\int_{0}^{\tau(\mathcal{T}^{\prime},\theta)}g_{\mathcal{T}^{\prime}}\circ\phi_{t}^{\mathcal{T^{\prime}}}(\bar{0},\theta)dt=\bar{G}_{\mathcal{T}^{\prime}}(0,\theta).$

Observe that for any $z\in\partial\mathcal{S}^{\prime}$ we can write

$\bar{G}_{\mathcal{T}^{\prime}}(z,\theta)=\bar{G}_{\varphi_{z}(\mathcal{T}^{\prime})}(0,\theta)=\bar{H}(\varphi_{z}(\mathcal{T}^{\prime}),\theta),$

so that $\beta(\bar{G}_{\mathcal{T}^{\prime}})=\bar{\mu}(\bar{H}).$ Using (42)
we obtain that for any $T>0$, up to an error of size
$M\|g_{\mathcal{T}^{\prime}}\|_{(1)}$,

$\begin{split}\int_{0}^{T}g_{\mathcal{T}^{\prime}}\circ\phi_{t}^{\mathcal{T^{\prime}}}(\bar{0},\theta)dt&=\int_{0}^{T}h\circ\bar{\phi}_{t}(\mathcal{T}^{\prime},\theta)dt=\sum_{i=0}^{k(T)-1}\int_{0}^{\tau(f^{i}(\mathcal{T}^{\prime},\theta))}h\circ\bar{\phi}_{t}(f^{i}(\mathcal{T}^{\prime},\theta))dt\\\
&=\sum_{i=0}^{k(T)-1}\bar{H}(f^{i}(\mathcal{T}^{\prime},\theta))\end{split}$

where $k(T)$ denotes the number of collisions of
$(\mathcal{T}^{\prime},\theta)$ with the boundary $\partial\Sigma$ in time
$T$. Finally, by the Birkhoff ergodic theorem applied to $\bar{H}$ and $\tau$,
we obtain for $\bar{\mu}$-almost every
$(\mathcal{T}^{\prime},\theta)\in\partial\Sigma\times[-\pi/2,\pi/2],$

$\begin{split}&\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}g_{\mathcal{T}^{\prime}}\circ\phi_{t}^{\mathcal{T^{\prime}}}(\bar{0},\theta)dt=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}h\circ\bar{\phi}_{t}(\mathcal{T}^{\prime},\theta)dt\\\
&\hskip
72.26999pt=\lim_{T\to\infty}\frac{k(T)}{T}\frac{1}{k(T)}\sum_{i=0}^{k(T)-1}\bar{H}(f^{i}(\mathcal{T}^{\prime},\theta))=\bar{\tau}^{-1}\bar{\mu}(\bar{H})=\bar{\tau}^{-1}\beta(\bar{G}_{\mathcal{T}^{\prime}}).\end{split}$

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