# Coding of billiards in hyperbolic 3-space

Pradeep Singh
Department of Mathematics, Indian Institute of Technology Delhi,
Hauz Khas, New Delhi 110016, INDIA

(July 22, 2021)

###### Abstract

In this article, we produce the coding rules of billiards for a class of ideal
polyhedrons in the 3-dimensional hyperbolic space. Building on it, further we
establish conjugacy between the space of pointed billiard trajectories and the
corresponding shift space of codes. This opens up a direct route to study the
related geometric properties via the analytical tools available in Symbolic
Dynamics.

_keywords:_ polyhedral billiards, pointed geodesics, Hausdorff metric,
hyperbolic 3-space, space of all subshifts.

2020 Mathematical Subject Classification  37B10, 37D40, 37D50

##  1 Introduction

The study of billiards is ubiquitous in physics with ambient space being
mainly euclidean. The demands of gravity naturally make us incline towards
asking similar questions in the non-euclidean spaces. Here we do the same in
the 3-dimensional hyperbolic space. Due to the lack of an embedding in
3-dimensional euclidean space, we make use of a suitable geometric model.

In this article, we investigate the coding of billiards in a class of
polyhedrons in the 3-dimensional hyperbolic space. Our polyhedrons sit in the
Poincaré ball model
$\mathbb{B}^{3}=\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}<1\\}$ with
vertices lying on the boundary $\partial\mathbb{B}^{3}$. Thus for such a class
of polyhedrons all vertices sit at infinity under the hyperbolic distance. We
will call this class _ideal_. For a fixed ideal polyhedron, we define a
billiard trajectory by letting a point inside the polyhedron run its course
with uniform speed along geodesics until it hits a face of the polyhedron,
wherein it reflects elastically observing the laws of specular reflection in
the hyperbolic plane defined by the incident geodesic and the normal geodesic
at the point of hit. After the hit it follows the reflected geodesic till
another face of polyhedron comes in its way. The process is repeated
infinitely in future. The same can be said about its past by imagining the
point running in reverse direction. Combining the past and future, we get a
bi-infinite collection of geodesic segments, which we label as a _billiard
trajectory_. It is to be noted here that we don’t take into account those
points which hit an edge or a vertex of the polyhedron in the past or the
future. By large, we are interested in non-truncated past and future. We label
the faces of the polyhedron arbitrarily with symbols. Isolating a billiard
trajectory, we notice that it naturally generates a bi-infinite sequence of
symbols that comprises of the symbols tumbling out as the point moves inside
the polyhedron hitting the faces turn wise. We pick a geodesic segment from
the billiard trajectory and call it the base of the _pointed billiard
trajectory_. Analogously, we get a _base symbol_ in the corresponding bi-
sequence of symbols. We call this new entity the _code for the pointed
billiard trajectory_. The main problem that we answer here is the
classification of the pointed billiard trajectories for the class of ideal
polyhedrons via certain grammar rules on the associated shift space.

In section 2, we describe the basic elements that are needed to deal with the
problem while describing the billiard map for the billiards in the hyperbolic
space. The section 3.1 introduces the concept of pointed geodesics for
billiards in ideal polyhedrons. We establish a metric on the space of pointed
geodesics which is topologically equivalent to the Hausdorff metric. In
section 3.2, we prove the main result which establishes the coding rules for
the billiard trajectories for a class of ideal polyhedrons in the hyperbolic
space. Using it, we prove the conjugacy between the space of pointed geodesics
and the associated symbolic space.

The coding of billiards in the hyperbolic plane has been studied in [3, 7, 8,
14]. While the problem in 2-dimensions is well supplemented by the geometric
intuition one gets from the two dimensional models of the hyperbolic plane,
the case of higher dimensions is devoid of anything of that sort. Due to this,
the work in 3-dimensions and higher is limited to the problem involving
relativistic billiards that has been dealt in [4, 5, 6, 15].

##  2 Preliminaries

In this section we lay down basic notions to be used further.

###  2.1 Hyperbolic 3-space

The hyperbolic 3-space is a 3-dimensional Riemannian manifold with a constant
negative curvature (sectional). It is a natural extension of the more commonly
appearing hyperbolic plane to three dimensions. The hyperbolic plane cannot be
embedded in $\mathbb{R}^{2}$ and thereby is studied using various models, most
common being the Poincaré half plane model denoted $\mathbb{H}^{2}$ and the
Poincaré disc model denoted $\mathbb{D}^{2}$, see e.g., [1]. Same holds true
for the hyperbolic 3-space which is dealt with two commonly used models -
Poincaré half space model denoted $\mathbb{H}^{3}$ and Poincaré ball model
denoted $\mathbb{B}^{3}$.

The Poincaré half space model is defined with the underlying space as
$\mathbb{H}^{3}=\\{(x,y,z)\in\mathbb{R}^{3}:z>0\\}$ and an attached metric
given by $ds^{2}=\dfrac{dx^{2}+dy^{2}+dz^{2}}{z^{2}}$. We think of the
boundary of $\mathbb{H}^{3}$ as the complex plane $\mathbb{C}$ with the
natural embedding and a point at infinity. This allows us to extract the
isometry group of $\mathbb{H}^{3}$ in terms of the available isometry group of
$\mathbb{C}$. If $\gamma:[a,b]\rightarrow\mathbb{H}^{3}$ is a path in
$\mathbb{H}^{3}$ given by $\gamma(t)=(x(t),y(t),z(t))$ with $t\in[a,b]$, then
$l_{\mathbb{H}^{3}}(\gamma)=\int_{\gamma}\dfrac{\sqrt{dx^{2}+dy^{2}+dz^{2}}}{z}$
gives its length. The distance between two points $A$ and $B$ is given by
$d_{\mathbb{H}^{3}}(A,B)=inf(l_{\mathbb{H}^{3}}(\gamma))$ with the infimum
running over all the paths originating from $A$ and terminating at $B$.

The underlying space for Poincaré ball model is
$\mathbb{B}^{3}=\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}<1\\}$ with
metric given by
$ds^{2}=\frac{4(dx^{2}+dy^{2}+dz^{2})}{(1-(x^{2}+y^{2}+z^{2}))^{2}}.$ If
$\gamma:[a,b]\rightarrow\mathbb{B}^{3}$ is a path in $\mathbb{B}^{3}$ that is
given by $\gamma(t)=(x(t),y(t),z(t))$, then its length is given by
$l_{\mathbb{B}^{3}}(\gamma)=\int_{\gamma}\dfrac{2\sqrt{dx^{2}+dy^{2}+dz^{2}}}{1-(x^{2}+y^{2}+z^{2})}.$
With points $A,B\in\mathbb{B}^{3}$, their distance is
$d_{\mathbb{B}^{3}}(A,B)=inf(l_{\mathbb{B}^{3}}(\gamma))$ with the infimum
running over all the paths originating from $A$ and terminating at $B$. We can
establish isometries between $\mathbb{H}^{3}$ and $\mathbb{B}^{3}$ analogous
to the Cayley map between $\mathbb{H}^{2}$ and $\mathbb{D}^{2}$. For a more
detailed account on the hyperbolic space, [2, 12] are suggested and for an
intuitive understanding of the same, [17] is recommended.

With the notion of distance on a geometric space, the question of _straight
lines_ or _geodesics_ arises naturally. Roughly, these are the locally
distance minimising curves of the space. For $\mathbb{H}^{3}$, the geodesics
are the euclidean straight lines perpendicular to the $\\{z=0\\}$ plane and
the euclidean semicircles hitting the $\\{z=0\\}$ plane orthogonally. The
boundary of $\mathbb{H}^{3}$ denoted $\partial{\mathbb{H}^{3}}$ comprises of
the plane $\\{z=0\\}$ and a point at infinity. The objects analogous to planes
in Euclidean geometry in case of Hyperbolic geometry are called _hyperbolic
planes_. A hyperbolic plane in $\mathbb{H}^{3}$ model is either a euclidean
plane hitting perpendicularly at the $\\{z=0\\}$ plane or a euclidean
hemisphere with its center lying on $\\{z=0\\}$ plane.

For $\mathbb{B}^{3}$, the geodesics are the euclidean straight lines passing
through origin and the portions of euclidean circles lying in $\mathbb{B}^{3}$
and hitting $\partial{\mathbb{B}^{3}}$ orthogonally. The boundary of
$\mathbb{B}^{3}$ comprises of the euclidean unit sphere centered at the
origin. A hyperbolic plane in $\mathbb{B}^{3}$ model is either part of a
euclidean plane passing through the origin lying in $\mathbb{B}^{3}$ or part
of a euclidean hemisphere inside $\mathbb{B}^{3}$ with its center lying on
$\partial{\mathbb{B}^{3}}$.

A _convex_ subset _A_ of the hyperbolic space is the one for which the closed
line segment $l_{xy}$, joining $x$ to $y$ is contained in _A_ for each pair of
distinct points $x$ and $y$ in _A_. All hyperbolic lines, hyperbolic rays,
hyperbolic line segments, hyperbolic planes, connected subsets of hyperbolic
planes are convex. Consider a hyperbolic plane $P$, the complement of $P$ in
the hyperbolic space has two connected components, which are the two _open
half spheres determined by P_. We refer to $P$ as the _boundary plane_ for the
half spheres it determines. Open half spheres and their closures are also
convex in hyperbolic space.

Let $S=\\{S_{\alpha}\\}_{\alpha\in\Lambda}$ be a collection of half spaces in
the hyperbolic space and for each $\alpha\in\Lambda$, let $P_{\alpha}$ be the
bounding plane for $S_{\alpha}$. The collection $S$ is _locally finite_ if for
each point $a$ in the hyperbolic space, $\exists\ \epsilon>0$ such that only
finitely many bounding planes $P_{\alpha}$ intersect the open hyperbolic ball
$U_{\epsilon}(a)$. A _hyperbolic polyhedron_ is a closed convex set in the
hyperbolic space that can be represented as the intersection of a locally
finite collection of closed half spaces. With this definition some
_degenerate_ objects still enter the picture. Simplest example that we can
take is that of any hyperbolic plane $P$ which satisfies the criteria but has
empty interior. To disallow this we must ensure that a hyperbolic polyhedron
is _nondegenerate_ which are the ones with nonempty interior. Here we will
consider only nondegenerate hyperbolic polyhedrons. Suppose $\Pi$ be a
hyperbolic polyhedron and $P$ a hyperbolic plane such that $\Pi$ intersects
$P$ and $\Pi$ is contained in a closed half space determined by $P$. If the
intersection $\Pi\cap P$ ends up in a point, we call it a $vertex$ of $\Pi$.
If $\Pi\cap P$ is a hyperbolic line or a hyperbolic line segment, it is called
an $edge$ of $\Pi$. If $\Pi\cap P$ is $P$ itself or its closed 2-dimensional
connected subset, then it is called a $face$ of $\Pi$.

Let $\Pi$ be a hyperbolic polyhedron and $e$ be an edge of $\Pi$ that is the
intersection of two faces $F_{1}$ and $F_{2}$ of $\Pi$. Let $P_{i}$ be the
hyperbolic plane containing $F_{i}\ \forall\ i=1,2$. Then $P_{1}\cup P_{2}$
divides the hyperbolic space into four connected components, out of which one
contains $\Pi$. The _interior angle_ of $\Pi$ at $e$ is the angle between
$P_{1}$ and $P_{2}$ measured in the component of $P_{1}\cup P_{2}$ containing
$\Pi$. $\Pi$ has an _ideal vertex_ at a vertex $v$ if there are two adjacent
faces of $\Pi$ that share $v$ as an endpoint on the boundary of the hyperbolic
space. A finite-faced polyhedron $\Pi$ in the hyperbolic space is called
$reasonable$ if it does not contain an open half-space. A _hyperbolic
k-hedron_ is a reasonable hyperbolic polyhedron with $k$ faces. A _compact
polyhedron_ is a hyperbolic polyhedron whose all vertices are in the
hyperbolic space. For $k\geq 4$, an _ideal k-hedron_ is a reasonable
hyperbolic polyhedron $\Pi$ that has $k$ faces with all the vertices lying on
the boundary of the hyperbolic space. A polyhedron is $isogonal$ if all its
vertices are of same type i.e., each vertex is surrounded by the same number
of faces and has same solid angle. Some features of an ideal polyhedron call
for attention. The primary one being that the faces of an ideal polyhedron are
themselves ideal polygons on which the billiard codes have been studied in [3,
7, 14]. An $k$-faced polyhedron has $(k+4)/2$ vertices whose surface can be
divided into $k$ faces consisting of ideal triangles. In particular, ideal
polyhedrons come only as even-faced. Since area of each ideal triangle is
$\pi$, the total surface area of the polyhedron itself comes out to be $k\pi$.

_This allows one to pose an interesting question of the relationship between
the space of codes corresponding to the billiards on faces of a polyhedron
with the space of codes for the bodily billiards that we seek to study in this
article_.

Secondly, not all euclidean polyhedrons have their ideal hyperbolic
counterparts. A necessary condition for the existence of an ideal polyhedra
corresponding to a euclidean polyhedra is if its vertices can be
simultaneously put on a circumscribing sphere. If two ideal polyhedrons have
the same number of vertices then they have the same surface area.

Figure 1: An ideal polyhedron in $\mathbb{H}$

All plane face angles and solid angles at the vertices of an ideal polyhedron
are zero. The angles between the faces (dihedral angles) are nonzero with
their supplementary counterparts summing up to $2\pi$. Thus for an ideal
isogonal tetrahedron, the dihedral angles come out to be $\pi/3$. In this
article, we are primarily focused on ideal isogonal polyhedrons. Figure 1
depicts a typical ideal tetrahedron. For further details on low-dimensional
geometry, [2, 12, 16, 17] are recommended. [10] provides for a classical
discussion on the related theory of fields.

###  2.2 Rolling out the Billiard Table

We will consider the Poincaré ball model $\mathbb{B}^{3}$ of the hyperbolic
space and carry out our investigations with the class of ideal hyperbolic
polyhedrons which we call simply _ideal polyhedrons_.

Let $\Pi$ be a $k$-faced ideal polyhedron in $\mathbb{B}^{3}$. A billiard
trajectory in the polyhedron $\Pi$ is a directed geodesic flow between each
pair of consecutive specular bounces of the boundary(not containing the
vertices and edges) of $\Pi$. We parameterize the boundary of the Poincaré
ball $\mathbb{B}^{3}$ using the azimuthal angle(longitude) denoted $\theta$
and the polar angle(latitude) denoted $\phi$ by considering it as a subset of
$\mathbb{R}^{3}$. Thus, we can represent a directed geodesic by the pair
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$, where
$(\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})$ are the intercepts made by a
directed geodesic on $\partial\mathbb{B}^{3}$ with the direction being from
$(\theta_{i},\phi_{i})$ to $(\theta_{f},\phi_{f})$. In this setting we get a
metric on $\partial\mathbb{B}^{3}$ which records the _great-circle distance_
between two points of $\partial{\mathbb{B}^{3}}$ in terms of latitudes and
longitudes.

The bounce map describes the evolution of geodesic arcs upon reflection from
the faces and is describes as follows: the pair
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ remains constant between any
two consecutive bounces from the faces and is changed to a new pair
$((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))$
at each bounce. The relationship between the incident and the reflected
geodesic arcs is given by
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\mapsto((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))=T((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})).$
We label $T$ as the bounce map. Suppose a geodesic arc
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ hits a point $a$ on a face $F$
of $\Pi$. Since $F$ is an orientable surface, we pick the inward unit normal
$\eta_{a}$ to $F$ at $a$ and select the unique geodesic emanating from $a$
lying in the plane containing the incident ray
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ and $\eta_{a}$, which is the
reflection of the incident ray in the mirror $\eta_{a}$. The reflected ray
$((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))$
takes the direction that goes away from the point $a$. A billiard trajectory
is a curve parameterized by the arc-length consisting of the geodesic arcs
which are reflected by the faces of the polyhedron $\Pi$. Therefore, a
trajectory can be expressed as
$\gamma=((\theta^{n}_{i},\phi^{n}_{i}),(\theta^{n}_{f},\phi^{n}_{f}))_{n\in\mathbb{Z}}$
where
$((\theta^{n}_{i},\phi^{n}_{i}),(\theta^{n}_{f},\phi^{n}_{f}))=T((\theta^{n-1}_{i},\phi^{n-1}_{i}),(\theta^{n-1}_{f},\phi^{n-1}_{f}))$.
We will not consider the billiard trajectories hitting the vertices or edges
of $\Pi$.

###  2.3 Tessellating the Hyperbolic Space and the Unfolding of Billiard
Trajectories

We can pick any one of $\mathbb{H}^{3}$ or $\mathbb{B}^{3}$ as a model for the
hyperbolic space. A _tessellation_ of $\mathbb{B}^{3}$ is a subdivision of
$\mathbb{B}^{3}$ into polyhedron tiles $\Pi_{i},\ i\in\Lambda$ obeying certain
rules. Firstly, $\forall\ a\in\mathbb{B}^{3},\ \exists\ i\in\Lambda$ with
$a\in\Pi_{i}$. Secondly, $\forall\ i\neq j,\ \Pi_{i}\cap\Pi_{j}$ is either
empty, a single vertex common to both, a single common edge or an entire
common face. Lastly, $\forall\ i\neq j,\ \exists$ an isometry $f_{i,j}$ of
$\mathbb{B}^{3}$ such that $f_{i,j}(\Pi_{i})=\Pi_{j}$. The definition for
$\mathbb{H}^{3}$ follows on same lines.

Let $\Pi$ be an ideal polyhedron in the hyperbolic space. We can reflect it
across any of its faces and the same can be done for the reflections and so
on. The collection of all such ideal polyhedrons with $\Pi$ generates a
tessellation of $\mathbb{B}^{3}$. This method is commonly known as _Katok-
Zemlyakov unfolding method_ and was given by A.B.Katok and A.N.Zemlyakov [9].
In case of a tetrahedron, its faces can be labeled as $1,2,3,4$ in an
arbitrary order. On reflecting about a face $i$, the labels on the
corresponding faces change to $1^{i},2^{i},3^{i},4^{i}$. This terminology is
retained in the same way for the ensuing reflections. The details of the
unfolding technique in the context of hyperbolic plane is described in [14].
This converts an ideal polyhedron in a hyperbolic space into a ‘tube’ of
isometric polyhedrons for a given trajectory in which the trajectory appears
as a full geodesic. Thus a billiard trajectory in a polyhedron $\Pi$ can be
_unfolded_ intuitively.

When describing the trajectories of particles over time in a typical dynamical
system, we often look at the state space at fixed intervals of time. This
discretisation is studied under Symbolic Dynamics. Some symbols are assigned
to the states which gives us a sequence that represents the trajectory
discreetly. Although lot of information is lost while discretising the state
space, still lot can be said about the trajectories while simplifying the
analysis tools. Through this we are interested in the long time behaviour of
the trajectories.

Formally, we fix a finite set $\mathcal{A}$ of symbols called the $alphabet$.
We define the $full\ \mathcal{A}\ shift$ as the space of all bi-infinite
sequences of symbols from $\mathcal{A}$. It’s denoted by
$\mathcal{A}^{\mathbb{Z}}=\\{x\ =\ ...x_{-1}.x_{0}x_{1}...:\ x_{i}\ \in\
\mathcal{A}\ \forall\ i\in\mathbb{Z}\\}$ with the product topology given by
the metric $d(x,y)=\inf\left\\{\frac{1}{2^{m}}:x_{n}=y_{n}\ \text{for}\
|n|<m\right\\},$ for two sequences $x=...x_{-1}.x_{0}x_{1}...$ and
$y=...y_{-1}.y_{0}y_{1}...\in\mathcal{A}^{\mathbb{Z}}$. The dynamics on the
full shift is launched by the $shift\ map\ \sigma$ defined by
$(\sigma(x))_{i}\ =\ x_{i+1}.$ A _shift space_ is a closed and invariant set
$X\subseteq\mathcal{A}^{\mathbb{Z}}$. For a more detailed account on this
subject, [11, 13] are recommended.

##  3 Pointed Geodesics and Billiards in an Ideal polyhedron

The concept of pointed geodesics was first introduced in [14] in the context
of billiards inside certain classes of polygons in the hyperbolic plane. Here,
we extend these definitions to the hyperbolic space.

###  3.1 Pointed Geodesics

######  Definition 3.1.

Let
$\gamma=((\theta^{n}_{i},\phi^{n}_{i}),(\theta^{n}_{f},\phi^{n}_{f}))_{n\in\mathbb{Z}}$
be a billiard trajectory in a polyhedron $\Pi$ in $\mathbb{B}^{3}$. Then for a
fixed $n\in\mathbb{Z}$, we call
$((\theta^{n}_{i},\phi^{n}_{i}),(\theta^{n}_{f},\phi^{n}_{f}))$ a base arc of
$\gamma$.

Note that every base arc is a compact subset of $\mathbb{B}^{3}$ and in turn
uniquely determines the billiard trajectory under the restrictions imposed by
the reflection rules.

######  Definition 3.2.

For a given base arc $((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ defining
$\gamma$, we will call $(\gamma,((\theta_{i},\phi_{i}),\\\
(\theta_{f},\phi_{f})))$ a _pointed geodesic_.

This implies that a pointed geodesic
$(\gamma,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})))$ is identified with

$\ldots(T^{-1}((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))).((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))(T((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})))\ldots\in\mathcal{K}(\mathbb{B}^{3})^{\mathbb{Z}}$

by fixing the position of the base arc
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ in the bi-sequence. We label
the faces of $\Pi$ with letters _1,…,k_ arbitrarily. A pointed geodesic now
can be encoded naturally by collecting the labels in the order in which it
hits the faces of the polyhedron $\Pi$, by marking the face hit by the base
arc and then reading the past and future hits. , Thus every pointed geodesic
produces a bi-infinite sequence $...a_{-1}.a_{0}a_{1}...$ with $a_{j}\ \in\
\\{1,...,k\\}$.

######  Definition 3.3.

We call
$\mathbf{G}=\mathbf{G}_{\Pi}=\\{\big{(}\gamma,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\big{)}:\gamma=\big{(}T^{n}((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\big{)}_{n\in\mathbb{Z}}\\}$,
the _space of all pointed geodesics_ on $\Pi$.

Here, $T^{0}((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ simply denotes
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$.
$\mathbf{G}\subsetneq\mathcal{K}({\mathbb{B}^{3}})$ and so ${\mathbf{G}}$ can
be endowed with the natural Hausdorff metric $d_{H}$, and in turn inherits the
Hausdorff topology.

We define a function $d_{\mathbf{G}}:\mathbf{G}\times\mathbf{G}\to\mathbb{R}$
by:

$\displaystyle
d_{\mathbf{G}}\Big{(}\big{(}\gamma,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\big{)},$
$\displaystyle\big{(}\gamma^{\prime},((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))\big{)}\Big{)}$
(1)
$\displaystyle=\max\\{d_{\partial\mathbb{B}^{3}}((\theta_{i},\phi_{i}),(\theta^{\prime}_{i},\phi^{\prime}_{i})),d_{\partial\mathbb{B}^{3}}((\theta_{f},\phi_{f}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))\\}.$

######  Proposition 3.1.

Suppose that $\mathbf{G}$ is the space of pointed geodesics associated with a
polyhedron $\Pi$ in $\mathbb{B}^{3}$, then $d_{\mathbf{G}}$ is a metric on
${\mathbf{G}}$.

###### Proof.

By definition, $d_{\mathbf{G}}$ is non-negative.
$d_{\mathbf{G}}\Big{(}\big{(}\gamma,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\big{)},\big{(}\gamma^{\prime},((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))\big{)}\Big{)}\\\
=0,$ implies
$d_{\partial\mathbb{B}^{3}}((\theta_{i},\phi_{i}),(\theta^{\prime}_{i},\phi^{\prime}_{i}))=0,d_{\partial\mathbb{B}^{3}}((\theta_{f},\phi_{f}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))=0.$
This means that
$(\theta_{i},\phi_{i})=(\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta_{f},\phi_{f})=(\theta^{\prime}_{f},\phi^{\prime}_{f})$,
which further gives
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))=((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))$.
If the base arcs corresponding to two pointed geodesics are coincident, then
their associated trajectories are also same due to the constraints of the
reflection rules. Thus,
$\Big{(}\gamma,\big{(}(\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})\big{)}\Big{)}\\\
=\Big{(}\gamma^{\prime},\big{(}(\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})\big{)}\Big{)}$.
The triangle inequality and symmetry for $d_{\mathbf{G}}$ follows from the
respective properties of $d_{\partial\mathbb{B}^{3}}$ which establishes
$d_{\mathbf{G}}$ as a metric on ${\mathbf{G}}$.∎

Next, we will show that the Hausdorff topology on $\mathbf{G}$ is same as the
topology on ${\mathbf{G}}$ given by $d_{\mathbf{G}}$. The metric $d_{H}$ on
$\mathbf{G}$ can be given as follows:

$\displaystyle
d_{H}\Big{(}\big{(}\gamma,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\big{)},\big{(}\gamma^{\prime},((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))\big{)}\Big{)}:=\
d_{H}\Big{(}\big{(}(\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})\big{)},\big{(}(\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})\big{)}\Big{)}$
(2)
$\displaystyle=\max\\{\sup_{Q\in\big{(}(\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})\big{)}}d\big{(}Q,\big{(}(\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})\big{)}\big{)},\sup_{Q\in\big{(}(\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})\big{)}}d\big{(}Q,\big{(}(\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})\big{)}\big{)}\\}$

This definition holds because $\gamma$ is described by its base arc uniquely.
If $\gamma\neq\gamma^{\prime}$ then
$d_{H}\Big{(}\big{(}\gamma,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\big{)},\big{(}\gamma^{\prime},((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))\big{)}\Big{)}>0$.
This notion of distance between two pointed geodesics is the Hausdorff
distance between the corresponding base arcs. We will represent the space of
all base arcs on a polyhedron $\Pi$ associated with billiard trajectories by
$\mathbf{B}(\Pi)$ or $\mathbf{B}$. As was the case in $2$-dimensions,
$\mathbf{B}\subset\mathcal{K}(\mathbb{B}^{3})$, where $\mathcal{K}(X)$ denotes
the space of all compact subsets of $X$, equipped with the Hausdorff topology.
Therefore, again in this case, the Vietoris topology and Hausdorff topology
match on $\mathbf{B}$ giving us a natural isometry between
$(\mathbf{G},d_{H})$ and $(\mathbf{B},d_{H})$ for any ideal polyhedron $\Pi$.
This gives us a one-one correspondence between the Vietoris topology on
$\mathbf{B}$ and the topology generated by $d_{H}$ on $\mathbf{G}$ leading to
$(\mathbf{G},d_{\mathbf{G}})$ and $(\mathbf{B},d_{\mathbf{G}})$ being
isometric.

######  Theorem 3.1.

Suppose $\mathbf{G}$ be the space of pointed geodesics on a polyhedron $\Pi$
in $\mathbb{B}^{3}$, then $d_{\mathbf{G}}$ and $d_{H}$ topologically
consistent on $\mathbf{G}$.

###### Proof.

Figure 2: An $\epsilon$-ball about a pointed geodesic

It is sufficient to prove that $d_{\mathbf{G}}$ and $d_{H}$ are topologically
same on $\mathbf{B}$. The space $\mathbf{B}$ via the metric $d_{H}$ gets the
induced topology given by $\mathbf{B}\ \subset\
\mathcal{K}({\mathbb{B}^{3}})$. For $\epsilon>0$, take
$V=\big{\\{}((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})):d_{\mathbf{G}}\Big{(}((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})),\big{(}(\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\Big{)}<\epsilon\big{\\}}.$
implying
$d_{\partial\mathbb{B}^{3}}((\theta^{\prime}_{i},\phi^{\prime}_{i}),\\\
(\theta_{i},\phi_{i})),d_{\partial\mathbb{D}}((\theta^{\prime}_{f},\phi^{\prime}_{f}),(\theta_{f},\phi_{f}))<\epsilon.$
Without any loss of generality, we can assume that $\epsilon$ is small enough
such that the $\epsilon$-tube of the base arcs about
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))$ doesn’t contain any vertex or
edge of $\Pi$, as has been shown in figure 2. Take the open balls
$U_{1},U_{2},...,U_{n}$ in $\mathbb{B}^{3}$ such that
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\subset\cup_{i=1}^{n}U_{i},\
((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\cap U_{i}\neq\emptyset\ \forall\
i=1,...,n$ and each $U_{i}$ lying inside the $\epsilon$-tube. Now
$<U_{1},...,U_{n}>$ is open in $\mathcal{K}({\mathbb{B}^{3}})$, which means
that $\mathbf{B}\ \cap<U_{1},...,U_{n}>$ is open in $\mathbf{B}$ and is
completely inside the $\epsilon$-tube. Thus
$((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f}))\in\mathbf{B}\
\cap<U_{1},...,U_{n}>\ \subset V.$

Conversely, consider a basic open set $\mathbf{B}\ \cap<U_{1},...,U_{n}>$
containing a base arc $((\theta_{i},\phi_{i}),(\theta_{f},\\\ \phi_{f}))$.
Without any loss of generality, we can take $U_{i}^{\prime}s$ as the open
discs in $\mathbb{B}^{3}$. Set $W_{ij}=\\{p\in\mathbb{B}^{3}:p\in U_{i}\ \cap\
U_{j}\ \forall i,j\in\\{1,...,n\\},i\neq j\\}.$ where each $W_{ij}$ is either
$\emptyset$ or has two points. Let
$W_{0}=\\{p\in\mathbb{B}^{3}:p\in(U_{i}\cap(\partial\Pi)_{k})\cup(U_{i}\cap(\partial\Pi)_{k+1})\
\forall\ i=1,...,n\\}.$ $(\partial\Pi)_{k}$ denotes the face of the polyhedron
with label $k$. Set $W=(\cup_{i,j=1,i\neq j}^{n}W_{ij})\cup W_{0}$ and pick
$\delta<inf_{p\in
W}(d_{\partial\mathbb{B}^{3}}(p,((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})))).$
Therefore, the $\delta$-tube
$V=\\{((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})):d_{\mathbf{G}}(((\theta^{\prime}_{i},\phi^{\prime}_{i}),(\theta^{\prime}_{f},\phi^{\prime}_{f})),((\theta_{i},\phi_{i}),(\theta_{f},\phi_{f})))<\delta\\}$
lies in $\mathbf{B}\ \cap<U_{1},...,U_{n}>$ completely.

∎

###  3.2 Billiards in Ideal polyhedrons

To give structure to our proof for the main result(Theorem 3.3), we will split
it into three parts, wherein we start with a lemma from elementary metric
spaces. Then we will follow it up with establishing of coding rules(Theorem
3.2) and then the conjugacy between the space of codes and the space of
pointed geodesics(Theorem 3.3).

######  Lemma 3.1.

Let $x\in\mathbb{B}^{3}$ be a fixed point. Then every hyperbolic plane
$p\in\mathbb{B}^{3}$ which does not contain $x$ divides $\mathbb{B}^{3}$ into
two open half spaces $H^{+}_{p}$ and $H^{-}_{p}$ with $x\in H^{-}_{p}$.
Further, let $(p_{n})_{n\geq 0}$ be a sequence of hyperbolic planes with the
additional properties that $p_{n+1}\subset H^{+}_{p_{n}}$ for all $n$ and
$d(p_{n},x)\rightarrow\infty$. Then the halfspaces $H^{+}_{p_{n}}$ determine a
unique ideal point $\eta\in\partial\mathbb{B}^{3}$ and each geodesic
originating from a point $y\in H^{-}_{p_{0}}$ and ending in $\eta$ penetrates
successively once through each of the hyperbolic planes $(p_{n})_{n\geq 0}$.

######  Theorem 3.2.

For a fixed $k\in 2\mathbb{N}+2$, let $\Pi\subset\mathbb{B}^{3}$ be a
$k$-faced ideal polyhedron with labelling as follows: Mark the faces
$1,2,...,k$ in an arbitrary order and then the vertex defined by faces
$i_{1},...,i_{m}$ takes the label $i_{1}...i_{m}$ for each
$i_{j}\in\\{1,...,k\\}$. Let $\Omega_{ij}$ denote the interior angle between
the pair of faces of $\Pi$ labelled $i$ and $j\ \forall\ i\neq j$ and
$i,j\in\\{1,...,k\\}$. Further, assume that
$\lambda_{ij}=\pi/\Omega_{ij}\in\mathbb{N}$ for each $i\neq j$ and
$i,j\in\\{1,...,k\\}$. Then an equivalence class $[...a_{-1}.a_{0}a_{1}...]$
denoted $(a_{j})$ with $...a_{-1}.a_{0}a_{1}...\in\\{1,...,k\\}^{\mathbb{Z}}$
is in $S(\Pi)$, if and only if
(a) $a_{j}\neq a_{j+1}\forall j\in\mathbb{Z}$.
(b) $(a_{j})$ does not contain more than $\lambda_{ij}$ repetitions of symbols
$i$ and $j$ which are labels of two adjacent faces.
(c) $(a_{j})$ does not contain an infinitely repeated sequence of labels of
faces meeting at a vertex.
Further, every equivalence class of such bi-infinite sequences corresponds to
one and only one billiard trajectory.

###### Proof.

First we establish the necessity of (a), (b) and (c). (a) follows from the
fact that a billiard trajectory cannot hit the same hyperbolic plane twice
because a geodesic cannot intersect a hyperbolic plane more than once. For
(b), suppose $F_{j},F_{k}$ be two adjacent faces of $\Pi$. Using a suitable
isometry we can consider without loss of generality that these two faces are
represented by the vertical hyperbolic planes $\\{y=0\\}$ and $\\{ax+by=0\\}$
in $\mathbb{H}^{3}$ for fixed $a,b\in\mathbb{R}$ with $a^{2}+b^{2}\neq 0,a\neq
0$. Suppose (b) does not hold and a billiard trajectory not starting or ending
in the vertex at $\infty$ of $\Pi$ hits faces $F_{j},F_{k}$ more than
$\lambda_{jk}$ times. Each segment of the corresponding billiard trajectory
lying between $F_{j}$ and $F_{k}$ is part of a semicircle hitting $\\{z=0\\}$
plane orthogonally because it cannot be a straight line perpendicular to
$\\{z=0\\}$ otherwise it will hit the vertex at $\infty$. On unfolding this
part of the trajectory we get a part of an orthogonal semicircle. On
projecting this part onto the $x-y$ plane we see that the projection is
subtending an angle $\mu_{jk}\Omega_{jk}>\pi$ at origin which leads to a
contradiction since the projection is a euclidean straight line in $x-y$
plane.
For (c), let $F_{j},F_{k},F_{l}$ be three faces of $\Pi$ meeting at a vertex
$v$. Using a suitable isometry we can consider without loss of generality that
these three faces are represented by the vertical hyperbolic planes
$\\{y=0\\},\\{ax+by=0\\}$ and $\\{cx+dy=0\\}$ with the corresponding vertex at
$\infty$. Let us label $\\{y=0\\}$ by $1$, $\\{ax+by=0\\}$ by $2$ and
$\\{cx+dy=0\\}$ by $3$ without any loss of generality, as referred in figure
3. Suppose (c) does not hold i.e. there exists a billiard trajectory whose
code contains an infinite word $w$ with $w_{i}\in\\{1,2,3\\}\forall i$. On
unfolding the corresponding part of the billiard trajectory we get a part of
the geodesic which is euclidean semicircle and orthogonal to $x-y$ plane. (b)
ensures that the copies of $\Pi$ come out of any euclidean circle drawn on the
plane $\\{z=0\\}$. Now a geodesic in $\mathbb{H}^{3}$ can hit only finitely
many copies of $\Pi$ generated while unfolding, which contradicts the infinite
cardinality of $w$.

Figure 3: An illustration of a vertex at $\infty$ defined by three vertically
placed hyperbolic planes

Conversely, we choose a sequence $(x_{i})_{i\in\mathbb{Z}}$ which satisfies
(a), (b) and (c) after fixing an ideal polyhedron $\Pi$ in $\mathbb{B}^{3}$
with faces labeled $1,2,3,4$ in an arbitrary order. Thus the chosen sequence
$(x_{i})_{i\in\mathbb{Z}}$ adheres to the restrictions imposed by (a), (b) and
(c). We aim at creating an algorithm that starting from
$(x_{i})_{i\in\mathbb{Z}}$ generates the corresponding unique billiard
trajectory in $\Pi$. We start by fixing an arbitrary point $A$ inside $\Pi$.
The sequence $(x_{i})_{i\in\mathbb{Z}}$ dictates us how to unfold the
polyhedron $\Pi$ in $\mathbf{B}$. This procedure will produce copies of
$\Pi=\Pi^{0}$ which we will label as $\Pi^{(i)}$ which serves as the reflected
copy of $\Pi^{(i-1)}$ in the face labeled $x_{i}$ for $i\geq 1$ and the
reflected copy of $\Pi^{(i+1)}$ in the face labeled $x_{i}$ for $i\leq-1$.
Thus we get a bi-sequence $(\Pi^{(i)})_{i\in\mathbb{Z}}$ of isometric copies
of $\Pi$. We will label the bi-sequence of faces in which the reflections are
taking place as $(p_{j})_{j\in\mathbb{Z}}$. Note that $p_{j}$ is labeled
$x_{j}\forall j\in\mathbb{Z}$. The bi-sequence $(p_{j})_{j\in\mathbb{Z}}$
obeys the condition (1) of the lemma 3.1. Indeed, $A$ is not contained in any
face $p_{j}$ and $p_{j+1}\subset H^{+}_{p_{j}}\ \forall j$. Next, we establish
condition (2) of lemma 3.1. Let $\delta>0$ be the least of all distances
between all the non-adjacent planes of $\Pi$. Then $d(p_{j+1},A)\geq
d(p_{j+1},p_{j})+d(p_{j},A)\geq\delta+d(p_{j},A)$ for the non-adjacent planes
$p_{j},p_{j+1}$. Thus if we have infinitely many such pairs $(p_{j},p_{j+1})$
of non-adjacent planes for $j\geq 0$, we get $d(p_{j},A)\rightarrow\infty$
which ensures the condition (2) of 3.1. This will define a unique limit point
$\beta\in\partial\mathbb{B}$. The unique geodesic emanating from $\alpha$ and
ending in $\beta$ is the unfolded trajectory corresponding to the code
$(x_{i})_{i\in\mathbb{Z}}$. If the non-adjacent pairs of planes are not
infinite in either direction then we look for the triples
$(p_{j-1},p_{j},p_{j+1})$ of non-adjacent planes of $\Pi$.

∎

######  Theorem 3.3.

For $k\in 2\mathbb{N}+2$, let $\Pi\subset\mathbb{B}^{3}$ be an ideal
polyhedron with labelling as follows: Mark the faces $1,2,...,k$ in an
arbitrary order and then the vertex defined by faces $i_{1},...,i_{m}$ takes
the label $i_{1}...i_{m}$ for each $i_{j}\in\\{1,...,k\\}$. Let $\Omega_{ij}$
denote the interior angle between the pair of faces of $\Pi$ labelled $i$ and
$j\ \forall\ i\neq j$ and $i,j\in\\{1,...,k\\}$. Further, assume that
$\lambda_{ij}=\pi/\Omega_{ij}\in\mathbb{N}$ for each $i\neq j$ and
$i,j\in\\{1,...,k\\}$. Let $\mathbf{G}$ be the space of pointed geodesics on
$\Pi$ and $X$ the space of all bi-infinite sequences
$...a_{-1}.a_{0}a_{1}...\in\\{1,...,k\\}^{\mathbb{Z}}$ satisfying (a), (b) and
(c) from Theorem 3.2. Then $(\mathbf{G},\tau)\simeq(X,\sigma)$.

###### Proof.

Let us define $h:(\mathbf{G},\tau)\to(X,\sigma)$ by
$h\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}=...a_{T^{-1}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}.a_{(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}a_{T(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}...$
$h\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}=h\big{(}\gamma^{\prime},(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}\\\
\Rightarrow...a_{T^{-1}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}.a_{(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}a_{T(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}...=\\\
...a_{T^{-1}(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))}.a_{(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))}a_{T(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))}...\\\
\Rightarrow\big{(}a_{T^{n}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}\big{)}_{n\in\mathbb{Z}}=\big{(}a_{T^{n}(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))}\big{)}_{n\in\mathbb{Z}}$

From 3.2, we have
$(T^{n}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}_{n\in\mathbb{Z}}=(T^{n}(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}_{n\in\mathbb{Z}}$
and $\\\
a_{(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}=a_{(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))}$
which implies
$\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}=\big{(}\gamma^{\prime},(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}.$
This gives the injectivity of h. We get the surjectivity of $h$ again by using
theorem 3.2 as each $(a_{j})_{j\in\mathbb{Z}}\in S(\Pi)$ defines a unique
billiard trajectory $\gamma$, which in turn implies that with the
corresponding $...a_{-1}.a_{0}a_{1}...$, we get a unique base symbol $a_{0}$,
which defines a base arc
$(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))$ on $\gamma$, giving a
unique pointed geodesic in $\mathbf{G}$. We have
$h\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}\\\
=...a_{-1}.a_{0}a_{1}...$
$h\circ\tau\Big{(}\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}\Big{)}=h\bigg{(}\tau\Big{(}\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}\Big{)}\bigg{)}\\\
=h\Big{(}\big{(}\gamma,T(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}\Big{)}=h\Big{(}\big{(}\gamma,(({\theta}_{1i},{\phi}_{1i}),({\theta}_{1f},{\phi}_{1f}))\big{)}\Big{)}\\\
$ (Here $(({\theta}_{1i},{\phi}_{1i}),({\theta}_{1f},{\phi}_{1f}))$ is the
reflected geodesic arc corresponding to the incident ray
$(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f})))\\\ $
$=...a_{T^{-1}(({\theta}_{1i},{\phi}_{1i}),({\theta}_{1f},{\phi}_{1f}))}.a_{(({\theta}_{1i},{\phi}_{1i}),({\theta}_{1f},{\phi}_{1f}))}a_{T(({\theta}_{1i},{\phi}_{1i}),({\theta}_{1f},{\phi}_{1f}))}...\\\
=...a_{T^{-1}T(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}.a_{T(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}a_{TT(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}...\\\
=...a_{(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}.a_{T(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}a_{T^{2}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))}...\\\
=\sigma\Big{(}h\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}\Big{)}=\sigma\circ
h\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}$

$\implies$ $h\circ\tau=\sigma\circ h$, which implies $h$ is a homomorphism.

Let $U=[x_{-m}...x_{-1}.x_{0}...x_{m}]$ in $(X,\sigma)$ be an open set in
$\mathbb{G}$. For a pointed bi-sequence $x\in$ U, we have corresponding
$(x_{n})_{n\in\mathbb{Z}}$, which generates a billiard trajectory $\gamma$
using 3.2. By pointing out the base arc
$(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))$ corresponding to
symbol $x_{0}$, we get a pointed geodesic
$(\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f})))$. Let’s label
its pointed bi-sequence by $y=...y_{-1}.y_{0}y_{1}...$. Since $x$ and $y$
belong to same equivalence class, $\exists$ an $s$ such that
$y_{[s-m,s+m]}=x_{-m}...x_{-1}.x_{0}...x_{m}$. Therefore,
$(\gamma,T^{-s}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f})))$ has its
associated pointed billiard bi-sequence
$h(\gamma,T^{-s}(({\theta}_{i},{\phi}_{i}),\\\ ({\theta}_{f},{\phi}_{f})))\
\in U.$ We will construct $m$ future and past copies of $\Pi$ in
$\mathbb{B}^{3}$ by reflecting $\Pi$ in its faces under the order given by
$h(\gamma,T^{-s}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f})))\ \in
U.$ Label $T^{-s}(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))$ as
$(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))$.

Define $\delta_{1}$ as
$\delta_{1}=\displaystyle\min_{i\in\\{1,...,k\\}}\big{\\{}d_{\partial\mathbb{B}^{3}}(A^{m}_{i},(\theta^{\prime}_{f},\phi^{\prime}_{f})),d_{\partial\mathbb{B}^{3}}(A^{-m}_{i},(\theta_{i}^{\prime},\phi_{i}^{\prime}))\big{\\}}.$
Choose $\epsilon$ such that $0<\epsilon<\delta_{1}$. If
$\big{(}\gamma^{\prime},(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}\in
B_{\epsilon}\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)},$
then
$[h\big{(}\gamma^{\prime},(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}]_{[-m,m]}\\\
=\displaystyle x_{-m}...x_{-1}x_{0}...x_{m}.$ Thus
$h\big{(}\gamma^{\prime},(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}\in
U$ and this implies that
$h\Big{(}B_{\epsilon}\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),\\\
({\theta}_{f},{\phi}_{f}))\big{)}\Big{)}\subseteq U.$ Therefore, $h$ is
continuous.

Conversely, let
$V=B_{\epsilon}\big{(}\gamma,(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))\big{)}$
be open in $\mathbb{G}$. Thus
$\big{(}\gamma^{\prime},(({\theta^{\prime}}_{i},{\phi^{\prime}}_{i}),({\theta^{\prime}}_{f},{\phi^{\prime}}_{f}))\big{)}\\\
\in V$ if and only if
$d_{\partial\mathbb{D}}((\theta_{i},\phi_{i}),(\theta^{\prime}_{i},\phi^{\prime}_{i})),d_{\partial\mathbb{D}}((\theta_{f},\phi_{f}),(\theta^{\prime}_{f},\phi^{\prime}_{f}))<\epsilon.$
We can tesselate $\mathbb{B}^{3}$ with $\Pi$ and its copies generated by
reflecting $\Pi$ about its sides and doing the same for the reflected copies
along the unfolded geodesic generated by $\gamma$. Label the vertices of $\Pi$
arbitrarily by $A_{1},A_{2},....,A_{k}$ and the vertices of the $i^{th}$ copy
of $\Pi$ by $A^{i}_{1},A^{i}_{2},....,A^{i}_{k}$. Define $p$ to be the largest
positive integer such that $A^{i}_{1},A^{i}_{2},....,A^{i}_{k}$ are not in
$\epsilon$-ball about $(({\theta}_{i},{\phi}_{i}),({\theta}_{f},{\phi}_{f}))$
for $i=-p,-p+1,...,0,1,...,p.$ which means that
$h^{-1}([x_{-p}...x_{-1}x_{0}...x_{k}])\subseteq V$ and thereby $h^{-1}$ is
continuous. ∎

It is noted here that the space _X_ described above is not closed as its limit
points of type $w\overline{abc}$ and $\overline{abc}w$ where $a,b,c$ are
symbols appearing on faces meeting at a common vertex, do not lie in _X_.

$\displaystyle
X=\\{...x_{-1}.x_{0}x_{1}...\in\\{1,...,k\\}^{\mathbb{Z}}:x_{i}\neq x_{i+1}\
\forall\ i\ and\ ...x_{-1}.x_{0}x_{1}...\neq\ w\overline{abc},\
\overline{abc}w\ for\ any$ (3) $\displaystyle a,\ b,\ c\ \in\\{1,...,k\\}\
sharing\ the\ same\ vertex\ and\ word\ w,...x_{-1}.x_{0}x_{1}...\neq
w(ab)^{\mu_{ab}}w^{\prime},$ $\displaystyle\mu_{ab}>\lambda_{ab}\forall\
labels\ a,b\ sharing\ an\ edge\ and\ arbitrary\ w,w^{\prime}\\}$

Therefore, we go further and define the closure of _X_ in
$\\{1,...,k\\}^{\mathbb{Z}}$, labelling it $\tilde{X}$. We can split
$\tilde{X}$ as $X\cup X^{\prime}$, where $X^{\prime}$ is the set of all limit
points of _X_.

$\displaystyle\tilde{X}=\\{...x_{-1}.x_{0}x_{1}...\in\\{1,...,k\\}^{\mathbb{Z}}:x_{i}\neq
x_{i+1}\forall i,...x_{-1}.x_{0}x_{1}...\neq
w(ab)^{\mu_{ab}}w^{\prime},\mu_{ab}>\lambda_{ab}$ $\displaystyle\forall\
labels\ a,b\ sharing\ an\ edge\ and\ arbitrary\ w,w^{\prime}\\}$

Thereby, $\tilde{X}$ has a finite forbidden set and thus is an _SFT_. This
places _X_ densely inside $\tilde{X}$. $\tilde{X}$ being the completion of _X_
is also the compactification of _X_.

## Data Availability Statement

Data sharing is not applicable to this article as no new data was created or
analysed in this study.

## Acknowledgements

This work was completed with the support from CSIR, India under the SPM
fellowship. The author wishes to thank Prof. Anima Nagar for the discussions
and providing many useful insights into the problem.

## References

  * [1] James W.Anderson, Hyperbolic Geometry, Springer-Verlag London Limited (2005)
  * [2] Francis Bonahon, Low-Dimensional Geometry, American Mathematical Society (2009)
  * [3] Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg, Billiards in ideal hyperbolic polygons, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 893-908
  * [4] M.V. Deryabin, L.D. Pustyl’nikov, On Generalized Relativistic Billiards in External Force Fields, Letters in Mathematical Physics 63: 195-207 (2003)
  * [5] M.V. Deryabin, L.D. Pustyl’nikov, Generalized relativistic Billiards, Regular and Chaotic Dynamics, 8:3 (2003), 283–296
  * [6] M.V. Deryabin, L.D. Pustyl’nikov, Exponential attractors in generalized relativistic billiards, Communications in Math. Physics, 248 (2004), 527–552
  * [7] Marie-Joya Giannoni, Dennis Ullmo, Coding Chaotic billiards : I, Physica D: Nonlinear Phenomena, 41 (1990), 371-390.
  * [8] Marie-Joya Giannoni, Dennis Ullmo, Coding Chaotic billiards : II, Physica D: Nonlinear Phenomena, Volume 84 (1995), 329-356.
  * [9] A. Katok, A. Zemlyakov, Topological Transitivity of Billiards in Polygons, Math. Notes, 18 (1975), 760-764
  * [10] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Pergamon Press, Oxford (1962)
  * [11] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press (1995)
  * [12] A. Marden, Outer Circles, Cambridge University Press (2007)
  * [13] Marston Morse and Gustav Hedlund, Symbolic dynamics, American Journal of Mathematics 60 (1938), 815–866
  * [14] Anima Nagar and Pradeep Singh, Finiteness in Polygonal Billiards on Hyperbolic Plane, Arxiv(http://arxiv.org/abs/2002.08002)(accepted for publication in TMNA)
  * [15] L.D. Pustyl’nikov, The law of entropy increase and generalized billiards, Russian Math. Surveys 54 (1999), no. 3, 650–651.
  * [16] Loring W. Tu, An Introduction to Manifolds, Springer (2008)
  * [17] William P. Thurston, The geometry and topology of three-manifolds, Princeton lecture notes (1980)
  * [18] William P. Thurston, Three-dimensional geometry and topology. Vol.1, Princeton University press (1997)