Finiteness in Polygonal Billiards on Hyperbolic Plane
FFiniteness in Polygonal Billiards on Hyperbolic Plane
Anima Nagar and Pradeep SinghDepartment of Mathematics, Indian Institute of Technology Delhi,Hauz Khas, New Delhi 110016, INDIAApril 08, 2020
Abstract
J. Hadamard studied the geometric properties of geodesic flows on surfaces of negativecurvature, thus initiating “Symbolic Dynamics”.In this article, we follow the same geometric approach to study the geodesic trajectories ofbilliards in “rational polygons” on the hyperbolic plane. We particularly show that the billiarddynamics resulting thus are just ‘Subshifts of Finite Type’ or their dense subsets. We furthershow that ‘Subshifts of Finite Type’ play a central role in subshift dynamics and while discussingthe topological structure of the space of all subshifts, we demonstrate that they approximateany shift dynamics. keywords: hyperbolic plane, polygonal billiards, pointed geodesics, subshifts of finite type,Hausdorff metric, space of all subshifts. a r X i v : . [ m a t h . D S ] A p r ontents ‘Mathematical billiards’ describe the motion of a point mass in a domain with elastic reflectionsfrom the boundary, and occur naturally in many problems in science. The billiards problem hastypically been studied in planar domains.A billiard in a ‘domain’ Π in the Euclidean plane is defined as a dynamical system describedby the motion of a point-particle within Π along the straight lines with specular reflections fromthe boundary ∂ Π. A domain is generally taken to be a subset of the plane that is compact with apiecewise smooth boundary. We refer to the point-particle under consideration as the billiard ball ,the path followed within Π as the billiard trajectory and the respective domain is called the billiardtable . This simple to describe mathematical system captures the essence of ‘chaotic dynamics’.The resultant ‘dynamics’ is heavily dependent upon the nature of the boundary of the billiardtable and the position as well as the orientation of the onset of the billiard. The curvature of thepart of the boundary being hit decides whether two parallelly launched billiard balls will comeout parallel, grow apart or will cross their paths. The overall trajectory is also determined by therelative placement of the pieces of the boundary with respect to each other. In this setting, theproblem of billiards has been studied classically in [12, 21, 22, 23, 43]. Although this problem initself is still ripe with many interesting open problems [24, 39, 40].
Symbolic dynamics is the dynamical study of the shift automorphism on the space of bi-infinitesequences on some set of symbols. The beginning of symbolic dynamics can be traced back to
J. Hadamard [28], when in 1898 he studied geodesics on surfaces of negative curvature from a2eometric point of view [16]. Though a systematic study of symbolic dynamics is said to havestarted with the work of
Marston Morse and Gustav Hedlund [37].Hadamard can also be credited to first study
Hadamard’s billiards which also counts as thefirst example proved to be possessing deterministic chaos. Along the same period of time,
G. D.Birkhoff working under the ambit of relativity and discrete dynamical systems was quite inter-ested in billiards, while studying the three-body problem . Many initial results on such geodesic flowswere compiled by
Gustov Hedlund [29]. Further work on Hadamard’s billiards was carried for-ward by
M. Gutzwiller in 1980s, see [25, 26, 27]. The study of polygonal billiards in hyperbolicplane besides its theoretical interest, appears in ‘General Relativity’ in an extended form of polyhe-dral billiards in a hyperbolic space (an n-dimensional manifold with constant negative ‘Riemanniancurvature’), see [14, 27]. Another approach can be seen in [34].For planar billiards in the
Euclidean plane , it has been observed that the route of dealingwith ‘Symbolic Dynamics’ is inherently ambiguous as this setting lacks in providing a one-onecorrespondence between the billiard trajectories and the corresponding natural codes generated bycollecting the labels of the sides of polygons being hit in order.This problem can be handled better in the hyperbolic plane .A hyperbolic plane is defined as a 2-dimensional manifold with a constant negative Gaussiancurvature. Since the hyperbolic plane cannot be embedded in a 3-dimensional Euclidean space, weare forced to work with various models of the hyperbolic plane, most common being the
Poincar´ehalf plane model H and the Poincar´e disc model D . We prefer the disc model for our work as it isEuclidean compact and this lets us follow the trajectories to each boundary which is the essence oftraditional billiards. This is not the case with the half plane model.In this setting, we carry out our investigations with a class of polygons ( being a subset of D )whose boundary comprises of finitely many geodesic segments, is piecewise smooth and intersectsin vertices at angles either 0 or those that divide π into integer parts. This ‘rationality’ at verticeslying in D is necessitated by the general technique that we follow which intrinsically takes the tiling of billiard tables (i.e. tessalations of the ambient hyperbolic plane) into account. For thisclass of polygons as billiard tables, we show that the associated billiard trajectories correspond tocountably many bi-infinite sequences . The work on these lines was initiated in [17, 18] and laterpursued in [14].We achieve uniqueness in the ‘coding’ by taking a billiard trajectory and breaking it into several ‘pointed geodesics’ , each described by a ‘base arc’ . These ‘pointed geodesics’ can be regarded ascompact subsets of D and form a space endowed with the Hausdorff topology. Under this splittingprocedure, we establish a one-one correspondence between the space of ‘pointed geodesics’ andthe corresponding space of bi-infinite sequences which we call as the associated shift space . The‘dynamics’ associated with these billiards is independent of the position and orientation of thepolygon in D , that allows us to choose any one of the isometric images appearing in the tiling asthe ‘fundamental polygon’.We introduce a metric d G on the space G of pointed geodesics which turns out to be equivalentto the classical ‘Hausdorff metric’. Under this metric and the transformation τ provided by the3bounce map’(the map that describes the specular reflection of the billiard ball at each hit withthe boundary), we consider ( G , τ ) to be a topological dynamical system. Further, collecting thesymbols on the sides being hit by the billiard ball, we establish a ‘coding’ for the trajectory. Thus,we get a ‘shift invariant’ set X of codes. We prove that X is a ‘shift of finite type (SFT)’ that isnot a ‘full k-shift’ or its dense subset. We establish a conjugacy between ( G , τ ) and ( X, σ ) , therebyinstituting an explicit route for studying the geometrical properties of billiards on this class ofpolygons via the symbolic dynamics on the corresponding space of bi-infinite sequences.At this point we would like to mention that such representations of geodesic flows by symbolicsystems have also been studied by Roy Adler and
Leopold Flatto [3] and
Caroline Series [41, 42]. One could also look into [1] for a more recent exposition.The two dimensional geometry that we consider is similar to [3], that also looks into the relationbetween geodesic flows on a compact surface of constant negative curvature and their associatedsymbolic dynamics. Though, their approach is to look into geodesic flows on a compact surface S = D / Γ of genus g ≥ F is guaranteed to exist with 8 g − cutting sequences that theyrefer to are the pointed geodesics in our case, though our concept of polygon is very distinct fromthe fundamental domain that they consider. Moreover, in [3, 41, 42] the discussions follow on themeasure-theoretic lines whereas our constructions and proofs run on topological arguments.It is noted here that our current problem has a setting that is different from the setting in[3, 41, 42]. The differences arise on several counts: the foremost being that the ambient spaceconsidered in [3, 41, 42] is a compact surface of genus g ≥ D in the background, thus allowing the billiard trajectories to lie on a non-compact surface.In [3, 41, 42] the authors start by choosing a Fuschian group
Γ such that D / Γ is a compact surfaceof genus g ≥ F is guaranteed toexist with 8 g − D which can even have an odd number of sides and assume no identification of sidesof Π.In [3], the authors have studied connection between the Geodesic flows and interval maps.For ( z, v ) ∈ T D , let ( γ ( t )) t ∈ R be the arclength parameterization of the unique geodesic passingthrough z ∈ D with the tangent line at z in the direction v and satisfying γ ( ) = z and γ ′ ( ) = v . The Geodesic flow on T D is the map φ from R × T D into T D defined by φ ( t, ( z, v )) = ( γ ( t ) , γ ′ ( t )) .The geodesic flow on the quotient of T D by a Fuschian group Γ is defined in the natural wayvia the projection map π ∶ T D → T ( D / Γ ) . Thus, to link the continuous time geodesic flow withthe discrete interval maps, various reductions are required. This is done by choosing a suitable cross section and the corresponding cross section map . A cross section is roughly considered as asubset C of T ( D / Γ ) which intersects the geodesic flow repeatedly in past and future. There areusually multiple choices available for the cross section.In our problem we choose the cross section explicitly to be T ( ∂ Π ) or under the projection map,simply ∂ Π. Intuitively, the part of the billiard trajectory between two consecutive hits is renderedirrelevant. We lift the billiard trajectories to the corresponding pointed geodesics on route toestablishing conjucacy between the corresponding space of pointed geodesics and the associatedspace of codes.On the contrary in [3], the authors choose a cross section C for which a second reductionis possible, which is given by a one-dimensional factor map. Through these two reductions, the4igure 1: Geodesic flow on D relationship between the geodesic flows and interval maps is further discussed. They then relatethese interval maps with symbolic systems via the Markov partitions . This line of separationbetween the two is crucial as we are able to discuss the case of billiards even when a vertex ofpolygon Π goes to ∂ D whereas in [3] the inherent restriction of D / Γ being compact is maintainedthroughout. The existence of the fundamental region F is essentially based on this. Our model isfairly simple.The underlying constructions are different in nature and this results in a difference in the natureof the associated symbolic sequences that they obtain in [3] from what we obtain here.Our results here motivate us to look into the larger picture of shift dynamics. We study thespace of all closed shift invariant sets of bi-infinite sequences on finite symbols, given the Hausdorfftopology. We particularly demonstrate that ‘shifts of finite type (SFTs)’ form a dense subset ofsuch a space. This gives us convergence in subshifts and we further study the dynamical propertiesinduced by such a convergence.In Section 2. we discuss some basic theory thus introducing our definitions and notations. Westudy our billiard dynamics and derive the associated symbolic dynamics and their properties inSection 3. And in Section 4. we study convergence of the dynamics in subshifts and polygonalbilliards. 5 Preliminaries
In this section, we lay down some basic notions for later usage, thus establishing the notation wewill henceforth use.
A discrete dynamical system ( X, f ) consists of a continuous self-map f on a metric space ( X, d ) .The orbit of a point x ∈ X is the set O( x ) = { x, f ( x ) , f ( x ) , . . . } . Here f n stands for the n − foldself-composition of f . We note that orbits are invariant sets i.e. f (O( x )) ⊆ O( x ) for all x ∈ X .The basic study in dynamics is to study the asymptotic behaviour of orbits of all x ∈ X .The point x is periodic if there exists n ∈ N such that f n ( x ) = x . The orbit of a periodic point,which is a finite set, is called a periodic orbit . The set of periodic points of f in X is denoted by P ( f ) . The set of all limit points of O( x ) is called the omega-limit set of f at x , and written as ω f ( x ) = ω ( x ) . Omega-limit sets are closed invariant sets. An element x ∈ X is called a recurrentpoint for f if for some n k ↗ ∞ , f n k ( x ) → x , i.e. x ∈ ω ( x ) . The set of recurrent points of f in X isdenoted by R( f ) . An element x ∈ X is called a nonwandering point for f if for every open U ∋ x , ∃ n ∈ N such that f n ( U ) ∩ U ≠ ∅ . The set of all nonwandering points of f is denoted by Ω ( f ) . Thesystem ( X, f ) is said to be non wandering if X = Ω ( f ) .The system ( X, f ) is said to be point transitive if there is an x ∈ X such that O( x ) = X .These points with dense orbits are called transitive points .The system is called topologically transitive when for every pair of nonempty, open sets U, V ⊂ X ,there exists n ∈ N such that f n ( U ) ∩ V ≠ ∅ . Notice that f is surjective, and so a nonempty, open U implies f − n ( U ) is nonempty and open for every n ∈ N .These definitions of point transitivity and topological transitivity are equivalent on all perfect,compact metric spaces.Many times we represent these systems as ( X, x , f ) or ( X, x ) where O( x ) = X . Such systemsare then termed pointed systems or ambits .The system is minimal when every orbit is dense. ( X, f ) is called topologically mixing if for every pair V, W of nonempty open sets in X , there isa N > f n ( V ) ∩ W is nonempty for all n ≥ N .For U, V ⊆ X , let N ( U, V ) = { n ∈ N ∶ f n ( U ) ∩ V ≠ ∅} be the hitting time set . We say that • ( X, f ) is transitive if for every pair of nonempty open sets U, V ⊆ X , N ( U, V ) is nonempty. • ( X, f ) is mixing if for every pair of nonempty open sets U, V ⊆ X, we have that N ( U, V ) iscofinite.For compact space, topological entropy was defined by Adler in [2]. Let A ( X ) (or simply A when the context is clear) denote the class of all open covers of X and A f ( X ) (or simply A f whenthe context is clear) denote the class of all finite open covers of X . Suppose that X is a compact6opological space and φ ∶ X → X is a continuous mapping. Let α i ∈ A for i = , , ..., n . We definethe join n ⋁ i = α of the covers α i by the formula n ⋁ i = α = α ∨ α ∨ ... ∨ α n = { U ∩ ... ∩ U n ∶ U i ∈ α i , i = , ..., n } . We define N X ( α ) or simply N ( α ) as the number of sets in a subcover of α of minimal cardinality.Set h ( α, φ ) = lim n →∞ n log N ( n − ⋁ i = φ − i α ) and h ( φ ) = sup α ∈ A h ( α, φ ) . The quantity h ( φ ) is called the topological entropy of φ .An equivariant map π ∶ ( X , f ) → ( X , f ) is a continuous map π ∶ X → X such that f ○ π = π ○ f .In particular, the diagram X f ———→ X π (cid:215)(cid:215)(cid:215)(cid:214) (cid:215)(cid:215)(cid:215)(cid:214) π X f ———→ X commutes.When π is a homeomorphism we call it a conjugacy and say that ( X , f ) and ( X , f ) are conjugate . When π is surjective we call it a factor map and say that ( X , f ) is a factor of ( X , f ) .We note that the properties of entropy, transitivity and topologically mixing are preserved ontaking factors.We refer to [4, 19, 47] for more details on topological dynamics.In [30], the concept of entropy has been extended by Hofer to non-compact Hausdorff spaces.According to [30], for a non-compact space X and T ∶ X → X , the topological entropy h X ( T ) = h X ∗ ( T ∗ ) , where X ∗ is a compactification of X and T ∗ is the extension of T on X ∗ . We note that with thisdefinition, it has been shown in [30]:(1) h X ( T k ) = kh X ( T ) for each positive integer k ,(2) If Y ⊂ X is T − invariant , then h Y ( T ∣ Y ) ≤ h X ( T ) ,(3) For non-compact spaces X and Y and systems ( X, T ) and ( Y, S ) , if φ ∶ X → Y satisfies φ ○ T = S ○ φ then h Y ( S ) ≤ h X ( T ) .It has been seen that pseudo-orbits, or more formally (cid:15) -chains, are important tools for inves-tigating properties of discrete dynamical systems. They usually capture the recurrent and mixingbehaviors depicted by the systems, even though they are highly metric dependent properties.7or x, y ∈ X , an (cid:15) -chain (or (cid:15) - pseudo-orbit) from x to y is a sequence { x = x , x , . . . , x n = y } such that d ( f ( x i − ) , x i ) ≤ (cid:15) for i = , . . . , n . The length of the (cid:15) -chain { x , x , . . . , x n } is said to be n . A point x ∈ X is chain recurrent if for every (cid:15) >
0, there is an (cid:15) -chain from x to itself. ( X, f ) is chain recurrent if every point of X is chain recurrent. ( X, f ) is chain transitive if for every x, y ∈ X and every (cid:15) >
0, there is an (cid:15) -chain from x to y . ( X, f ) is chain mixing if for every (cid:15) > x, y ∈ X , there is an N > n ≥ N , there is an (cid:15) -chain from x to y of length exactly n . Theorem 2.1. [4] For ( X, f ) , ω ( x ) is chain transitive for all x ∈ X . Further, if ( X, f ) is chaintransitive then it can be embedded in a larger system where it is an omega limit set. Theorem 2.2. [7] If ( X, f ) is chain transitive, then either ( X, f ) is chain mixing or ( X, f ) factorsonto a non-trivial periodic orbit. We suggest the enthusiastic readers to look into [4, 7, 38] and the references therein for manyinteresting chain properties.
Shift spaces are built on a finite set A of symbols which we call the alphabet . Elements of A arecalled letters .We define the full A shift as the collection of all bi-infinite sequences or bi-infinite sequences ofsymbols from A . It is denoted by A Z = { x = ...x − .x x ... ∶ x i ∈ A ∀ i ∈ Z } . The product topology on A Z is metrizable and a compatible metric defined on it can be givenas: d ( x, y ) = inf { m ∶ x n = y n for ∣ n ∣ < m } , (1)for any two sequences x = ...x − .x x ... and y = ...y − .y y ... ∈ A Z .The shif t map σ on the full shift A Z maps a point x to the point σ ( x ) whose ith coordinate is ( σ ( x )) i = x i + . A shift space is a closed, invariant (i.e. σ ( X ) ⊆ X ) set X ⊆ A Z .Observe that for x, y ∈ X , ∃ m > d ( x, y ) < − m ⇔ ∃ k > x [− k,k ] = y [− k,k ] . For n ∈ N , (a nonempty) w ∈ A n is a word of length n , and we write ∣ w ∣ = n . If the word w is apart of the word v then we say that w is a subword of v and we write w ⊏ v .8imilarly for any x ∈ A Z , we write w ⊏ x if w appears in x as a block, i.e. w = x [ k,k + n ] = x k x k + . . . x k + n and for m ∈ N , we write the concatenation w m = w . . . w ·„„„„„„„‚„„„„„„„¶ m -times . The collection of allnonempty words in A Z is A ∗ = ⋃ n ∈ N A n .Let L( X ) ⊂ A ∗ be the language of shift space X i.e. the set of all nonempty words appearing inany x ∈ X . Let F ⊂ A ∗ be the set of blocks that never appear in any x ∈ X . Usually we can writethe shift space X as X F for some collection F of forbidden blocks over A , i.e. F ⊆ L( X ) c . Everyshift space can also be defined by its language X = X F = X L( X ) c . Notice that for X = A Z we have F = ∅ .For the shift space X ⊆ A Z , the system ( X, σ ) is called a subshift . ( X F , σ ) is called a subshift of finite type (SFT) if the list of forbidden words F can be taken tobe finite. If M + M − step SFT.Thus an M − step SFT has the property that if uv, vw ∈ L( X ) and ∣ v ∣ ≥ M +
1, then uvw ∈ L( X ) aswell.Every SFT is topologically conjugate to an edge SFT X A , presented by some square nonnegativematrix A . Here A is viewed as the adjacency matrix of some directed graph G , whose edge set isthe alphabet A of the SFT. X A ⊂ A Z is the space of bi-infinite sequences corresponding to walksthrough the graph G . Here for every i , the terminal vertex of x i equals the initial vertex of x i + .Thus, a SFT can also be denoted as X G or X A where G is the associated graph and A is thetransition matrix.A non-negative matrix A = ( a ij ) is called irreducible if for every i, j there is k ∈ N such that A kij >
0, and is called aperiodic if there is k ∈ N such that ( A k ) ij >
0, for every i, j .A subshift ( X, σ ) is transitive if for every pair of words u, v ∈ L( X ) , there exists a word w ∈ L( X ) with uwv ∈ L( X ) . ( X, σ ) is mixing if there exists an N ∈ N such that such a w can be choosenwith ∣ w ∣ = n for all n ≥ N .A SFT ( X A , σ ) is transitive if and only if the transition matrix A is irreducible, and is mixingif and only if the transition matrix A is aperiodic.SFTs can also be viewed as vertex shifts , where the vertices are labelled by elements in A ∗ and a directed edge connects two vertices if the concatenation of the labels of these vertices is apermissible word in the language. The transition matrix here is a Boolean matrix.Here codes play an important role. The most important codes for us are those that do notchange with time i.e. codes which intertwine with the shift ( σoφ = φoσ ) . A common example ofsuch codes is the sliding block code which we define as follows: Let X be a shift space over A . Let B be another alphabet and ¯ φ ∶ B m + n + ( X ) → B be a map called (m+n+1)-block map or simply block map , where B m + n + ( X ) is the set of all ( m + n + ) -blocks in L( X ) . We define a map φ ∶ X → B Z given by y = φ ( x ) where y i = ¯ φ ( x [ i − m,i + n ] ) . The map φ is called the sliding block code with memory m and anticipation n induced by ¯ φ . 9 heorem 2.3. (Curtis-Hedlund-Lyndon) Let ( X, σ X ) and ( Y, σ Y ) be subshifts over finite alphabets A and B respectively. A continuous map φ ∶ X → Y commutes with the shift (i.e., φoσ X = σ Y oφ )if and only if φ is a sliding block code. If a sliding block code φ ∶ X → Y is onto, it is called a factor code from X onto Y and Y iscalled a factor of X. If φ is one-one, then it is called embedding of X into Y. φ is called a conjugacy from X to Y, if it is invertible. The shift spaces in this case are called conjugate and we write X ≡ Y. Conjugacies carry n − periodic points to n − periodic points and in general preserve the dynamicalstructure.The topological entropy of a subshift X is given as h X = lim n →∞ log ∣ B n ∣ n , where ∣ B n ∣ denotes the number of words in L( X ) of length n . It is known that the topologicalentropy of an irreducible SFT X A equals log λ where λ is the Perron eigenvalue of A .We refer to [35] for more details.There is an interesting illustration of Theorem 2.1 for SFTs. Theorem 2.4. [9] Let Λ ⊂ X F be an invariant and closed subset. Then there is a point x ∈ X F such that Λ = ω ( x ) if and only if Λ is chain transitive. Remark 2.1.
One of the consequences of the above observation, also observed independently in[38] is - in the case of SFTs, chain transitivity is equivalent to transitivity . Lastly, we recall some characterization of the language of a subshift from [5] and build on it.We skip the trivial proofs since the arguments are similar to the ones given in [5]. Recall that as v varies over L( X ) , the cylinder sets [ v ] = { x ∈ X ∶ x [− k, − k +∣ v ∣] = v, k ∈ N } comprise a bases of clopen sets on X .(a) ( X, σ ) is transitive(irreducible) if and only if for all v ∈ L( X ) and all w ∈ L( X ) , there exists a ∈ L( X ) such that vaw ∈ L( X ) .(b) ( X, σ ) is minimal if and only if whenever v ∈ L( X ) then v ⊏ x for all x ∈ X .(c) ( X, σ ) is mixing if and only if whenever v, w ∈ L( X ) there exists N ∈ N such that for all k ∈ N there exists a k ∈ L( X ) with ∣ a k ∣ = N + k such that va k w ∈ L( X ) .(d) ( X, σ ) is non wandering if and only if for all v ∈ L( X ) there exists a ∈ L( X ) , such that vav ∈ L( X ) .(e) ( X, σ ) is chain recurrent if and only if for all v ∈ L( X ) , there exists a , a , . . . a n ∈ A and v , v , . . . v n − ∈ L( X ) with ∣ v ∣ = ∣ v ∣ = . . . = ∣ v n − ∣ , such that va v , v a v , . . . , v n − a n v ∈L( X ) . 10f) ( X, σ ) is chain transitive if and only if for all v ∈ L( X ) and all w ∈ L( X ) , there ex-ists a , a , . . . a n ∈ A and v , v , . . . v n − ∈ L( X ) with ∣ v ∣ = ∣ v ∣ = . . . = ∣ v n − ∣ , such that va v , v a v , . . . , v n − a n w ∈ L( X ) .(g) ( X, σ ) is chain mixing if and only if for all v ∈ L( X ) and all w ∈ L( X ) , there exists N ∈ N such that for all k ∈ N there exists a m j ∈ A for j = , . . . , N + k and v , v , . . . v N + k − ∈ L( X ) with ∣ v ∣ = ∣ v ∣ = . . . = ∣ v N + k − ∣ , such that va m v , v a m v , . . . , v N + k − a m N + k w ∈ L( X ) . For a metric space ( X, d ) , we denote as 2 X − the space of all nonempty closed subsets of X , and K( X ) − the space of all compact subsets of X , endowed with the Hausdorff topology. We note thatusually K( X ) ⊆ X but for compact X , 2 X = K( X ) .This has a natural induced metric.Given a point p ∈ X and a closed set A ⊆ X , recall d ( p, A ) = inf a ∈ A d ( p, a ) . On 2 X we define the Hausdorff metric : For A, B ∈ X d H ( A, B ) = max { sup a ∈ A d ( a, B ) , sup b ∈ B d ( b, A )} (2)We note that d H is a psuedo-metric on 2 X and a metric on K( X ) .For (cid:15) >
0, let A (cid:15) = { y ∈ X ∶ d ( y, a ) < (cid:15), a ∈ A } be the (cid:15) − neighbourhood of A . Thus, d H ( A, B ) < (cid:15) if and only if each set is in the open (cid:15) neighborhood of the other i.e. A ⊂ B (cid:15) and B ⊂ A (cid:15) , or,equivalently, each point of A is within (cid:15) of a point in B and vice-versa.When X is compact, we occasionally use an equivalent topology on 2 X . Define for any collection { U i ∶ ≤ i ≤ n } of open and nonempty subsets of X , ⟨ U , U , . . . U n ⟩ = { E ∈ X ∶ E ⊆ n ⋃ i = U i , E ⋂ U i ≠ φ, ≤ i ≤ n } (3)The topology on 2 X , generated by such collection as basis, is known as the Vietoris topology .If { A n } is a sequence of closed sets in a X then ⋃ n { A n } = ⋃ n { A n } ∪ lim sup n { A n } , where lim sup n { A n } = ⋂ k ⋃ n ≥ k { A n } . We recall,
Lemma 2.1.
For a metric space ( X, d ) , and { A n } a sequence in ( X , d H ) ,(a) If { A n } converges to A in K( X ) then ⋃ n { A n } is compact.(b) If ⋃ n { A n } is compact and { A n } is Cauchy then A n converges to lim sup { A n } .(c) If X is complete then X is complete.(d) If X is compact then X is compact. e) If X is separable then so is X .(f ) If i X ∶ X → X is given by x ↦ { x } , then i X is an isometric inclusion.(g) For f ∶ X → X a continuous map of metric spaces, there is induced the map f ∗ ∶ K( X ) →K( X ) defined by A ↦ f ( A ) . If f is uniformly continuous or continuous, then the map f ∗ is alsouniformly continuous or continuous, respectively.(h) The set of all finite subsets of X is dense in K( X ) . These results also hold when 2 X is given the Vietoris topology.We refer [6, 31, 36] for more details.The Gromov-Hausdorff metric furthers the idea of the Hausdorff metric. Given two compactmetric spaces X and Y , we define d GH ( X, Y ) = inf f,g d H ( f ( X ) , g ( Y )) (4)where f ( X ) , g ( Y ) denote an isometric embedding of X, Y into some metric space Z and theinfimum is taken over all such possible embeddings. Lemma 2.2. [20] The following hold with respect to the Gromov-Hausdorff metric:1. If
X, Y are compact metric spaces, the d GH ( X, Y ) < ∞ .2. If
X, Y are not compact then it is possible that d GH ( X, Y ) = , without X, Y being isometric.For example [ , ] , Q ∩ [ , ] .3. Metric spaces X i → X if and only if for every (cid:15) > , there exists (cid:15) ′ ≥ (cid:15) such that every (cid:15) ′ − netin X is a limit of (cid:15) − nets in X i .4. Compact X and Y are isometric if and only if d GH ( X, Y ) = . We denote the set of all isometric compact metric spaces endowed with the Gromov-Hausdorffmetric as M . Then the following is known about the metric space (M , d GH ) .1. (M , d GH ) is separable and complete.2. (M , d GH ) is not locally compact or compact.3. The set of all finite metric spaces is dense in (M , d GH ) .We refer to [13, 20] for more details.In a memoir [45](published after his death) P. Urysohn showed that there is a metric space ( U, d ) which has the following universality and homogeneity properties:(a) U is complete and separable.(b) Every separable metric space is isometric to a subspace of ( U, d ) .(c) Any isometry between finite subsets of U extends to an isometry of U .Furthermore, ( U, d ) is unique up to isometry.Such a universal metric space is called a Urysohn space . We refer to many articles of
AnatolyVershik for more details on such universal spaces without mentioning any references here.Later
Vladimir Uspenskij [46] proved that the Urysohn universal metric space U is homeo-morphic to the Hilbert space l . 12 .4 Geodesics and Polygons in the Hyperbolic Plane A hyperbolic space is a space that has a constant negative sectional curvature. We work in dimen-sion 2 and call it a hyperbolic 2-space or a hyperbolic plane. We will use two models of hyperbolicplane, namely, the Poincar´e half plane model which we denote as H and the Poincar´e disc modeldenoted as D . Both of them model the same geometry in the sense that they can be related by anisometric transformation that preserves all the geometrical properties. We refer to [8, 10, 11] formore details.The underlying space of the Poincar´e half plane model is the upper half-plane H in the complexplane C , defined to be H = { z ∈ C ∣ Im ( z ) > } . We use the usual notion of point and angle that H inherits from C . The metric on H is defined by ds = dx + dy y . If γ ∶ [ a, b ] → H is a path in H that is parametrised in [ a, b ] with γ ( t ) = x ( t ) + iy ( t ) , then the length l H ( γ ) of the path γ is defined by l H ( γ ) = ∫ ba √ dx + dy y dt. Given two points P,Q ∈ H , the distance d H ( P, Q ) = inf ( l H ( γ )) , where infimum is taken over all the paths from P to Q.The Poincar´e disk model is described by D = { z ∈ C ∶ ∣ z ∣ < } . The metric on D is defined by ds = ( dx + dy )( − ( x + y )) . The
Cayley transformation C ∶ H → D defined by z → ( z − i )/( z + i ) is a conformal isometry here.If γ ∶ [ a, b ] → D is a path in D that is parametrised in [ a, b ] with γ ( t ) = x ( t ) + iy ( t ) , then thelength l ( γ ) of the path γ is defined by l D ( γ ) = ∫ ba ¿``(cid:192) ( dx + dy )( − ( x + y )) dt. Given two points P,Q ∈ D , the distance d D ( P, Q ) = inf l D ( γ ) , straight lines as geodesics . They are the locally distance minimisingcurves of the space. Under the metric imposed on H , we get the geodesics to be the euclidean linesperpendicular to real axis and the euclidean semicircles which are orthogonal to the real axis. Wenote that the real axis along with the point at infinity gives the boundary ∂ H of H . In case of D with the above defined metric, we get the geodesics to be the euclidean lines passing through thecentre of the disc and the euclidean circles orthogonal to ∂ D , the boundary of D .A subset A of the hyperbolic plane is convex if for each pair of distinct points x and y in A ,the closed line segment l xy joining x to y is contained in A . Hyperbolic lines, hyperbolic rays, andhyperbolic segments are convex. Given a hyperbolic line l , the complement of l in the hyperbolicplane has two components, which are the two open half-planes determined by l . A closed half-planedetermined by l is the union of l with one of the two open half-planes determined by l . We refer to l as the bounding line for the half-planes it determines. Open half-planes and closed half-planes in H are convex.Let H = { H α } α ∈ Λ be a collection of half-planes in the hyperbolic plane, and for each α ∈ Λ,let l α be the bounding line for H α . The collection H is called locally finite if for each point z inthe hyperbolic plane, there exists some (cid:15) > l α of thehalf-planes in H intersect the open hyperbolic disc U (cid:15) ( z ) where U (cid:15) ( z ) = { w ∈ H ∶ d H ( z, w ) < (cid:15) } . A hyperbolic polygon is a closed convex set in the hyperbolic plane that can be expressed as theintersection of a locally finite collection of closed half-planes. Under this definition, there are somesubsets of the hyperbolic plane that satisfy this criteria, but we do not want them to be consideredas hyperbolic polygons. For example, a hyperbolic line l is a hyperbolic polygon, because it is aclosed convex set in the hyperbolic plane that can be expressed as the intersection of the two closedhalf-planes determined by l . A hyperbolic polygon is nondegenerate if it has nonempty interior elseit is called degenerate . We will work only with nondegenerate polygons here.Let P be a hyperbolic polygon and let l be a hyperbolic line so that P intersects l and so that P is contained in a closed half-plane determined by l . If the intersection P ∩ l is a point, we saythat this point is a vertex of P . The other possibilities are that the intersection P ∩ l is either aclosed hyperbolic line segment, a closed hyperbolic ray, or all of l . We call this intersection side of P . Let P be a hyperbolic polygon, and let v be a vertex of P that is the intersection of twosides s and s of P . Let l k be the hyperbolic line containing s k . The union l ∪ l divides thehyperbolic plane into four components, one of which contains P . The interior angle of P at v isthe angle between l and l , measured in the component of the complement of l ∪ l containing P .A hyperbolic polygon P in the hyperbolic plane has an ideal vertex at v if there are two adjacentsides of P that are either closed hyperbolic rays or hyperbolic lines that share v as an endpoint atinfinity.A finite-sided polygon P in the hyperbolic plane is called reasonable if P does not contain anopen half-plane. A hyperbolic n-gon is a reasonable hyperbolic polygon with n sides. A compactpolygon is a hyperbolic polygon whose all vertices are in the hyperbolic plane. A compact hyperbolicn-gon is regular if its sides have equal length and if its interior angles are equal. For each n ≥ ideal n-gon is a reasonable hyperbolic polygon P that has n sides and n vertices. Thus, anideal polygon is a hyperbolic polygon whose all vertices are ideal points (i.e. lying on the boundaryof the hyperbolic plane). The hyperbolic polygons with vertices lying both inside the hyperbolic14lane and on its boundary are called semi-ideal polygons . An angle at a vertex of a polygon in D is called rational if it is of the form π / n where n ∈ N and n >
1. The corresponding vertex is calleda rational vertex . Thus, a compact polygon is labeled rational if all its vertices are rational anda semi-ideal polygon is called rational if all its non-zero vertex angles are rational. We note thatthe angle at vertices that are ideal points is zero and so by definition ideal polygons are vacuousrational polygons.
Theorem 2.5. (Gauss-Bonnet formula) Let P be a hyperbolic polygon with vertices v , ..., v n . Foreach k , let α k be the interior angle at v k . Then area H ( P ) = ( n − ) π − Σ nk = α k . Therefore, an ideal polygon has infinite perimeter and finite area from Gauss-Bonnet formula.In particular, an ideal ( k + ) -sided polygon has an area kπ and thus is the largest possible polygon inhyperbolic plane. The compact polygons have finite perimeter and area strictly less than kπ, k ∈ N .The semi-ideal ones have infinite perimeter and area less than or equal to kπ, k ∈ N . The polygonsin the hyperbolic plane enjoy a very special feature, namely, the similar polygons are congruent.In particular, all ideal n-gons are congruent to each other. This feature allows us to work in asimplified situation of a symmetrically placed n-gon.We refer to [8, 44] for more details. We can consider any one of H or D as the model for the hyperbolic plane. A tessellation of D is asubdivision of D into polygonal tiles Π i , i ∈ Λ satisfying the following conditions:(1) ∀ z ∈ D , ∃ i ∈ Λ such that z ∈ Π i ,(2) ∀ i ≠ j, Π i ∩ Π j is either empty, or a single vertex common to both, or an entire commonedge,(3) ∀ i ≠ j, ∃ an isometry f i,j of hyperbolic plane such that f i,j ( Π i ) = Π j .Informally speaking, a collection of tiles tessellate the hyperbolic plane if they cover the plane,don’t overlap, and are of same shape and size.Let Π be an ideal polygon in the hyperbolic plane. Then, we can reflect it across each one of itssides and the same procedure can be applied to the reflections and so on. The collection of all suchideal polygons obtained, along with Π gives us a tessellation of D . The figure below shows one suchexample, where we start with an ideal triangle Π. Its sides are labeled as 1 , , i , the labels change to 1 i , i , i . This labelling proceeds in sameway for further reflections. Similar tessellation can be obtained for the rational compact polygonsand for the semi-ideal polygons with the vertex angles either 0 or rational.The unfolding technique that we discuss ahead was formally introduced to the domain of billiarddynamics by A.B.Katok and
A.N.Zemlyakov [32]. This method converts a polygon on a planeinto a surface on which the billiard trajectories appear as geodesics. Under this technique a billiardtrajectory in a polygon Π can be unfolded in an intuitive procedure as follows: Instead of reflectingthe trajectory in a side of Π, we reflect Π itself in that side which gives a copy of the reflected rayin the new polygon. The join of this new directed line segment with the incident ray in Π lies on aline. We say that the billiard trajectory has been unfolded at this hit point. When we apply thisprocedure to the whole billiard trajectory, it gives us straightened version of the billiard trajectory,which we call as unfolded billiard trajectory . 15igure 2: Tessellation of D Figure 3: Unfolding of a billiard trajectoryMore details on tessellations of polygons in the hyperbolic plane can be found in the elaboratedwork of S.Katok in [33], wherein it has been described via the Fuschian groups and the fundamentalregions. The tessellations are described via the action of a discrete and properly discontinuous groupon a fundamental region. Here, we avoid this nomenclature as our primary object of concern is thepolygon (which is already fixed to start with) and the billiard dynamics happening on it.We refer to [10, 15, 43] for more details.
We consider the Poincar´e disc model D of the hyperbolic plane.16e carry out our investigations with a class of hyperbolic polygons which we call as idealpolygons, compact rational polygons and semi-ideal rational polygons . The ideal polygons are theones with all the vertices on ∂ D and the compact rational polygons are the ones for which all thevertices lie in D and have angles that divide π into integer parts. The polygons that lie in ‘semi-idealrational’ class are the ones for which either the vertices lie on ∂ D or the vertex angles divide π intointeger parts.We follow the general construction as defined in [17] for the ideal polygons and in [18] forcompact rational polygons.Let Π be a k-sided polygon in D . A billiard trajectory in the polygon Π is a directed geodesic flowbetween each pair of consecutive specular bounces of the boundary(not containing the vertices) of Π.We have a simple choice for the coordinate system for the directed geodesic arcs. We parameterizethe boundary of the Poincar´e disc D using the azimuthal angle by considering it as a subset of C .Thus, we can represent a directed geodesic by the pair ( θ, φ ) , where θ, φ are the intercepts made bya directed geodesic on ∂ D with the direction being from θ to φ . In this setting we have a naturalmetric on ∂ D given by d ∂ D ( φ , φ ) = ∣ φ − φ ∣ . (5)The evolution of geodesic arcs during reflection is given by the bounce map defined as follows: the pair ( θ, φ ) remains constant between any two consecutive bounces and is changed to a newpair ( θ ′ , φ ′ ) at each bounce, where the relationship is given by ( θ, φ ) ↦ ( θ ′ , φ ′ ) = T ( θ, φ ) . We label T as the bounce map which is described as follows: tan ( φ ′ / ) = e − r cot ( φ / ) where r is rapidity defined by e r = cot ( ω / ) ,ω being half of the angle subtended by the reflecting side of Π at the centre of D . Define M = cot ( φ − θ )/ , then φ ′ = θ ′ + cot − M ′ where M ′ is given by M ′ = − M e r sin ( θ / )[ + e − r cot ( θ / )] − sinh ( r ) sin ( θ ) . A billiard trajectory is a curve that is parameterized by the arc-length and consists of the geodesicarcs which are reflected by the walls of the polygon Π. Therefore, a trajectory can be expressed as γ = {( θ n , φ n ) n ∈ Z } ( θ n , φ n ) = T ( θ n − , φ n − ) . We do not consider billiard trajectories starting or ending in vertices of Π.The billiard trajectories here follow the grammar rules defined and discussed by
Marie-JoyaGiannoni, Dennis Ullmo in [17, 18]. Based on that
Simon Castle, Norbert Peyerimhoff,Karl Friedrich Siburg [14] prove the rules for ideal polygons.
Theorem 2.6. [14] Let Π ⊂ D be an ideal polygon with counter-clockwise enumeration , ..., k . Anequivalence class [ ...a − .a a ... ] denoted ( a j ) with ...a − .a a ... ∈ { , ..., k } Z is in S ( Π ) if and onlyif (1) ( a j ) does not contain immediate repetitions, i.e., a j ≠ a j + ∀ j ∈ Z and(2) ( a j ) does not contain an infinitely repeated sequence of labels of two adjacent sides.Moreover, every equivalence class of pointed billiard sequences corresponds to one and only onebilliard trajectory. We refer [14, 17, 18] for further details on the geometric aspects of billiards in hyperbolic plane.
Definition 3.1.
Let γ = ( θ n , φ n ) n ∈ Z be a billiard trajectory in a polygon Π in D . For a fixed n ∈ Z ,we will call ( θ n , φ n ) as a base arc of the trajectory γ . We note that base arcs are compact subsets of D .A base arc uniquely determines the billiard trajectory under the restrictions imposed by thespecular reflection rule. Definition 3.2.
For the base arc ( θ, φ ) defining γ , we call ( γ, ( θ, φ )) a pointed geodesic . Thus a pointed geodesic ( γ, ( θ, φ )) is identified with the element . . . ( T − ( θ, φ )) . ( θ, φ )( T ( θ, φ )) . . . ∈ K( D ) Z by clearly establishing the position of the base arc ( θ, φ ) . A natural way of encoding a pointedgeodesic is to seize the order in which it hits the sides of Π, starting from the side hit by thebase arc and then reading the past and future hits of the trajectory and pointing out the symbolcorresponding to the base arc. If we label the sides of Π anti-clockwise from 1 to k, then everypointed geodesic produces a bi-infinite sequence ...a − .a a ... with a j ∈ { , ..., k } . Definition 3.3.
Define G = G Π = {( γ, ( θ, φ )) ∶ γ = ( T n ( θ, φ )) n ∈ Z } as the space of all pointed geodesics on Π . G ⊊ K( D ) and so G can be equipped with the natural Hausdorff metric d H , and so is endowedwith the Hausdorff topology. T ( θ, φ ) is simply written as ( θ, φ ) .We define a function d G ∶ G × G → R as follows: d G (( γ, ( θ, φ )) , ( γ ′ , ( θ ′ , φ ′ ))) = max { d ∂ D ( θ, θ ′ ) , d ∂ D ( φ, φ ′ )} . (6)where d ∂ D is as in Equation 5. Proposition 3.1.
Let G be the space of pointed geodesics on a polygon Π in D , then d G defines ametric on G .Proof. Clearly, d G is non-negative. If d G (( γ, ( θ, φ )) , ( γ ′ , ( θ ′ , φ ′ ))) = , then d ∂ D ( θ, θ ′ ) = , d ∂ D ( φ, φ ′ ) = . Therefore, θ = θ ′ , φ = φ ′ implying ( θ, φ ) = ( θ ′ , φ ′ ) . If base arcs of two pointed geodesics are samethen the corresponding trajectories are also same because of the dynamics provided by the bouncemap. Therefore, (( γ, ( θ, φ ))) = (( γ ′ , ( θ ′ , φ ′ ))) . The symmetry and triangle inequality for d G follows from the respective properties of d ∂ D (theboundary of the Poincar´e disc). Therefore, d G is a metric on G .We show that the Hausdorff topology on G is same as the topology on G given by d G . Recallthat d H on G can be given as follows: d H (( γ, ( θ, φ )) , ( γ ′ , ( θ ′ , φ ′ ))) ∶= d H (( θ, φ ) , ( θ ′ , φ ′ )) = max { sup Q ∈( θ,φ ) d ( Q, ( θ ′ , φ ′ )) , sup Q ∈( θ ′ ,φ ′ ) d ( Q, ( θ, φ ))} (7)Note that the above definition works because γ is uniquely determined by a base arc and if γ ≠ γ ′ then d H ( γ, γ ′ ) >
0. Thus, this notion of distance between two pointed geodesics is just theHausdorff distance between the corresponding base arcs. We denote the space of all base arcs ona polygon Π that are associated with billiard trajectories by B ( Π ) or simply B , when the contextis clear. Note that B ⊂ K( D ) and is a bounded subset of D when Π is compact. Thus, in thiscase, the Vietoris topology and Hausdorff topology are equivalent on B . We, thereby get a naturalisometry between ( G , d H ) and ( B , d H ) for a polygon Π, giving a one-one correspondence betweenthe Vietoris topology on B and the topology generated by d H on G . Under the same pretence, wealso have the d G metric on B and the natural isometry between ( G , d G ) and ( B , d G ) . Theorem 3.1.
Let G be the space of pointed geodesics on a polygon Π in D , then d G and d H generate the same topology on G . roof. With the above discussion, it is sufficient to prove that d G and d H generate the same topologyon B . The topology on B given by the metric d H is the induced topology on B ⊂ K( D ) . For (cid:15) > V = {( θ ′ , φ ′ ) ∶ d G (( θ ′ , φ ′ ) , ( θ, φ )) < (cid:15) } . Therefore, d ∂ D ( θ, θ ′ ) , d ∂ D ( φ, φ ′ ) < (cid:15). Without any loss of generality, we assume that (cid:15) is small enough such that the (cid:15) -tube of the basearcs about ( θ, φ ) doesn’t contain any vertex of Π. Consider the open balls U , U , ..., U n in D suchthat ( θ, φ ) ⊂ ∪ ni = U i , ( θ, φ ) ∩ U i ≠ ∅ ∀ i = , ..., n and each U i lying inside the (cid:15) -tube. Since < U , ..., U n > is open in K( D ) , therefore B ∩ < U , ..., U n > is open in B and is lying in the (cid:15) -tube. Therefore, we have ( θ, φ ) ∈ B ∩ < U , ..., U n > ⊂ V. Conversely, consider a basic open set B ∩ < U , ..., U n > containing a base arc ( θ, φ ) . Withoutthe loss of generality, we assume that U ′ i s are open discs in D . Define W ij = { p ∈ D ∶ p ∈ U i ∩ U j ∀ i, j ∈ { , ..., n } , i ≠ j } . Note that each W ij is either ∅ or contains two points. Define W = { p ∈ D ∶ p ∈ ( U i ∩ ( ∂ Π ) k ) ∪ ( U i ∩ ( ∂ Π ) k + ) ∀ i = , ..., n } . ( ∂ Π ) k denotes the side of the polygon with label k . Define W = (∪ ni,j = ,i ≠ j W ij ) ∪ W and choose δ < inf p ∈ W ( d ∂ D ( p, ( θ, φ )) . Then, the δ -tube V = {( θ ′ , φ ′ ) ∶ d G (( θ ′ , φ ′ ) , ( θ, φ )) < δ } lies inside B ∩ < U , ..., U n > . Theorem 2.6 ensures that the set S ( Π ) is not dependent on the choice of the ideal polygon Π. Theelements of S ( Π ) are restricted only by the rules (1.)and (2.) defined there.Let Π an ideal polygon that is symmetrically placed on D , and let G be the space of pointedgeodesics on Π.Define a map τ ∶ G → G with its action on G described as follows : τ (( γ, ( θ, φ ))) = ( γ, T ( θ, φ )) ∀ j ∈ Z . We will study the dynamics of ( G , τ ) under the metric d G . Theorem 3.2.
Let Π ⊂ D be an ideal polygon with counter − clockwise enumeration , ..., k and G be the space of pointed geodesics on Π . Suppose X be the space of all pointed bi-infinitesequences ...a − .a a ... ∈ { , ....k } Z satisfying the rules: a j ≠ a j + ∀ j ∈ Z and(2) ...a − .a a ... does not contain an infinitely repeated sequence or bi-infinite sequence of labels oftwo adjacent sides.Then ( G , τ ) ≃ ( X, σ ) .Proof. Define h ∶ ( G , τ ) → ( X, σ ) by h ( γ, ( θ, φ )) = ...a T − ( θ,φ ) .a ( θ,φ ) a T ( θ,φ ) ...h ( γ, ( θ, φ )) = h ( γ ′ , ( θ ′ , φ ′ ))⇒ ...a T − ( θ,φ ) .a ( θ,φ ) a T ( θ,φ ) ... = ...a T − ( θ ′ ,φ ′ ) .a ( θ ′ ,φ ′ ) a T ( θ ′ ,φ ′ ) ... ⇒ ( a T n ( θ,φ ) ) n ∈ Z = ( a T n ( θ ′ ,φ ′ ) ) n ∈ Z From [14] , we see that ( T n ( θ, φ )) n ∈ Z = ( T n ( θ ′ , φ ′ )) n ∈ Z and a ( θ,φ ) = a ( θ ′ .φ ′ ) ⇒ ( γ, ( θ, φ )) = ( γ ′ , ( θ ′ , φ ′ )) . This gives the injectivity of h.The surjectivity of h is established from the fact that each ( a j ) j ∈ Z ∈ X defines a unique billiardtrajectory γ . Therefore with corresponding ...a − .a a ... , we get a unique base symbol a , whichfurther picks a base arc ( θ, φ ) on γ , thereby giving us a unique pointed geodesic in G i.e. h ( γ, ( θ, φ )) = ...a − .a a ...h ○ τ (( γ, ( θ, φ ))) = h ( τ (( γ, ( θ, φ ))))= h (( γ, T ( θ, φ )))= h (( γ, ( θ , φ )))= ...a T − ( θ ,φ ) .a ( θ ,φ ) a T ( θ ,φ ) ... = ...a T − T ( θ,φ ) .a T ( θ,φ ) a T T ( θ,φ ) ... = ...a ( θ,φ ) .a T ( θ,φ ) a T ( θ,φ ) ... = σ ( h ( γ, ( θ, φ )))= σ ○ h ( γ, ( θ, φ )) Therefore, h ○ τ = σ ○ h , implying that h is a homomorphism.Consider an open set V = B (cid:15) ( γ, ( θ, φ )) in G . Thus ( γ ′ , ( θ ′ , φ ′ )) ∈ V if and only if d ∂ D ( θ, θ ′ ) , d ∂ D ( φ, φ ′ ) < (cid:15). D with Π and its copies generated by reflecting Π about its sides and doing the same forthe reflected copies along the unfolded geodesic generated by γ . Let us label the vertices of Π inanticlockwise sense by A , A , ...., A k and the vertices of the i th copy of Π by A i , A i , ...., A ik . Definep to be the largest positive integer such that A i , A i , ...., A ik /∈ ( θ − (cid:15), θ + (cid:15) ) ⨉( φ − (cid:15), φ + (cid:15) ) ∀ i = − p, − p + , ..., , , ..., p. Then h − ([ x − p ...x − x ...x k ]) ⊆ V. Therefore, h − is continuous.Figure 4: (cid:15) -tube about an unfolded billiard trajectoryConsider an open set U = [ x − m ...x − .x ...x m ] in ( X, σ ) . Pick an arbitrary pointed bi-sequence x ∈ U. Then from corresponding ( x n ) n ∈ Z , we get abilliard trajectory γ using 2.6. By pointing out the base arc ( θ, φ ) corresponding to symbol x , weget a pointed geodesic ( γ, ( θ, φ )) whose billiard bi-sequence, we label as y = ( y i ) . Now, in general y may not be in U , but since x and y belong to same equivalence class, ∃ an s such that y [ s − m,s + m ] equals x − m ...x − .x ...x m . Therefore, ( γ, T − s ( θ, φ )) has its associated pointed billiard bi-sequence h ( γ, T − s ( θ, φ )) ∈ U. We construct m future and m past copies of Π in D by reflecting Π about its sides as suggested by h ( γ, T − s ( θ, φ )) ∈ U. Label T − s ( θ, φ ) as ( θ ′ , φ ′ ) .Let δ be defined as follows : δ = min i ∈{ ,...,k } { d ∂ D ( A mi , φ ′ ) , d ∂ D ( A − mi , θ ′ )} . (cid:15) such that 0 < (cid:15) < δ . If ( γ ′ , ( θ ′ , φ ′ )) ∈ B (cid:15) ( γ, ( θ, φ )) , then [ h ( γ ′ , ( θ ′ , φ ′ ))] [− m,m ] = x − m ...x − x ...x m . Thus h ( γ ′ , ( θ ′ , φ ′ )) ∈ U, i.e. h ( B (cid:15) ( γ, ( θ, φ ))) ⊆ U. Therefore, h is continuous.Thus, the space of all bi-infinite sequences on a k-sided ideal polygon is given by X = { ...x − .x x ... ∈ { , , ..., k } Z ∶ x i ≠ x i + ∀ i and ...x − .x x ... ≠ ab, wab, abw f or any a, b ∈{ , ..., k } and word w } . Therefore, X is not closed as the limit points of X of type ab , wab , abw do not lie in X . Thus,we further look for the closure of X in { , ..., k } Z and define ˜ X = X ∪ X ′ where X ′ is the set of alllimit points of X . Hence, ˜ X = { ...x − .x x ... ∈ { , ..., k } Z ∶ x i ≠ x i + ∀ i } and thereby is an SFT with forbidden set { , , ..., kk } . Thus, X is a dense subset of an SFT.We notice that ˜ X is a completion of X , therefore it is also a compactification of X .Thus, here we obtain a pair of conjugacies π between ( G , τ ) and ( B , T ) given by π ∶ G → B suchthat π ( γ, ( θ, φ )) = ( θ, φ ) and h between ( G , τ ) and ( X, σ ) with h defined as above.In particular, the diagram X σ ———→ X h − (cid:215)(cid:215)(cid:215)(cid:214) (cid:215)(cid:215)(cid:215)(cid:214) h − G τ ———→ G π (cid:215)(cid:215)(cid:215)(cid:214) (cid:215)(cid:215)(cid:215)(cid:214) π B T ———→ B commutes. For an ideal polygon with k vertices, the closure of the space of pointed bi-infinite sequences hasforbidden set F = { , , ..., kk } . A = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ . . .
11 0 1 . . . ... . . . ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ . Its eigenvalues turn out to be − , − , ..., − , k − B n ( ˜ X ) = k ( k − ) n . This implies that h ˜ X = lim n →∞ log B n n = lim n →∞ log ( k ( k − ) n ) n = k − . Since X is a dense subset of ˜ X , they share the same entropy [30].We also note that the matrix A is aperiodic and so ( ˜ X, σ ) is mixing. Since every element of X ′ consists as subword the infinite strings of the form ab for a, b ∈ { , . . . , k } , we note that ( X, σ ) isalso mixing.Thus ( G , τ ) is a mixing system with topological entropy k −
1, with k being the number of idealvertices of the polygon Π. Now, we consider the case when the vertices of Π lie inside D . The coding for most boundedsystems is non-exact and so is the case here for the compact polygons except for the ones thattile D . Therefore, to ensure that the fundamental domain tesselates the disc, we only consider thepolygons with rational angles. Recall that, an angle at a vertex of a polygon in D is called rational if it is of the form π / n where n ∈ N and n >
1. We will call the corresponding vertex a rationalvertex . First, we will establish the coding rules for such rational polygons motivated by the discussion in[18]. Theorem 3.3 has been originally discussed in [18]. Here, we present an alternative approachand proof for the same using the basic techniques of topological dynamical systems. We followthis by establishing the conjugacy between the space of corresponding pointed geodesics and thesymbolic space defined under the coding rules presented in Theorem 3.3.
Theorem 3.3.
Let Π ⊂ D be a compact rational polygon with anti-clockwise enumeration of sideslabeled , , ..., k . Label the vertices of Π as v , ..., v k with Ω , ..., Ω k being the respective interiorangles such that the adjacent sides of v i are i and i + . Further, assume that λ i = π / Ω i ∈ N for each i ∈ { , ..., k } . Then an equivalence class of bi-infinite sequences, ( a j ) with ...a − .a a ... ∈ { , ..., k } Z is in S ( Π ) (the space of all equivalence classes of bi-infinite sequences of Π ) if and only if ...a − .a a ... does not contain any immediate repetitions of symbols i.e., a j ≠ a j + ∀ j ∈ Z .(2) ...a − .a a ... does not contain more than λ i repetitions of two successive symbols i and i + forevery i ∈ { , ..., k } .Moreover, every equivalence class of such bi-infinite sequences corresponds to one and only onebilliard trajectory.Proof. First, we will establish the necessity of (1) and (2). Consider a bi-infinite sequence ...a − .a a ... attached to a billiard trajectory. In D , two distinct geodesics can have at most one intersection,therefore (1) holds. Now, suppose (2) does not hold for ...a − .a a ... i.e., there exists a subword w of ...a − .a a ... with µ i (> λ i ) repetitions of letters i and i +
1. Without any loss of generality, let usassume that w starts with i . Then, on unfolding the corresponding part of the trajectory µ i times,we get the condition Ω i µ i > π at the vertex v i , which gives a contradiction.Now, suppose ...a − .a a ... be a bi-infinite sequence for which (1) and (2) hold. We will constructa unique billiard trajectory defined by it in Π. We will split our proof into three parts. First we willprove that if we consider a bi-sequence satisfying (1) and (2) which is periodic, it uniquely definesa billiard trajectory. Then, we will show that the set of periodic bi-infinite sequences satisfying (1)and (2) is dense in the set of all bi-infinite sequences satisfying (1) and (2). Lastly, this will allowus to construct a unique billiard trajectory against an arbitrary bi-sequence satisfying (1) and (2)as the limit of a sequence comprising of periodic bi-infinite sequences satisfying (1) and (2).The set of all periodic bi-infinite sequences is dense in a full k -shift. If we remove bi-infinitesequences containing words ( i i + ) µ i and ( i + i ) µ i where µ i > λ i = π / Ω i ∀ i , and the ones for which a i = a i + for any i , the set of all periodic bi-infinite sequences in the remaining space is still dense.Indeed, if we start with ...a − .a a ... satisfying (1) and (2), we can define a sequence of periodicpoints as follows: take x = ( a − .a a ) ∞ , x = ( a − a − .a a a ) ∞ and so on. An issue with a typical term of such a sequence can be that the maximal repeating wordmay have starting and ending letters same or adjacent which violates (1) and (2) respectively. Asa remedy to this, whenever such a violation happens, we drop that term from the sequence. Thisprocedure still leaves a subsequence (because of the fact that ...a − .a a ... itself satisfies (2) whichcomprises of periodic points and converges to ...a − .a a ... . Thus, we have the required densenessproperty.Suppose ...a − .a a ... is an arbitrary bi-sequence obeying (1) and (2). Let ( x m ) m ∈ N be a sequencegenerated by the above construction, then x m → ...a − .a a ... . Now, each x m has a unique geodesic γ m associated with it. Since, D is geodesically complete, the sequence of geodesics, ( γ m ) m ∈ N converges to a limit geodesic γ , which acts as the unique geodesic associated with ...a − .a a ... . Onfolding γ back into the fundamental polygon Π, we get a unique billiard trajectory associated with [ ...a − .a a ... ] . Theorem 3.4.
Let Π ⊂ D be a compact rational polygon with anti-clockwise enumeration of sideslabeled , , ..., k . Label the vertices of Π as v ,..., v k with Ω ,..., Ω k being the respective interiorangles such that the adjacent sides of v i are i and i + . Further, assume that λ i = π / Ω i ∈ N foreach i ∈ { , ..., k } . Let G be the space of pointed geodesics on Π and X the space of all bi-infinitesequences ...a − .a a ... ∈ { , ..., k } Z satisfying (1) and (2) from Theorem 3.3. Then ( G , τ ) ≃ ( X, σ ) .Proof. Define h ∶ ( G , τ ) → ( X, σ ) by h ( γ, ( θ, φ )) = ...a T − ( θ,φ ) .a ( θ,φ ) a T ( θ,φ ) ... ( γ, ( θ, φ )) = h ( γ ′ , ( θ ′ , φ ′ ))⇒ ...a T − ( θ,φ ) .a ( θ,φ ) a T ( θ,φ ) ... = ...a T − ( θ ′ ,φ ′ ) .a ( θ ′ ,φ ′ ) a T ( θ ′ ,φ ′ ) ... ⇒ ( a T n ( θ,φ ) ) n ∈ Z = ( a T n ( θ ′ ,φ ′ ) ) n ∈ Z From Theorem 3.3 , we see that ( T n ( θ, φ )) n ∈ Z = ( T n ( θ ′ , φ ′ )) n ∈ Z and a ( θ,φ ) = a ( θ ′ .φ ′ ) ⇒ ( γ, ( θ, φ )) = ( γ ′ , ( θ ′ , φ ′ )) . This gives the injectivity of h.The surjectivity of h is established from the fact that each ( a j ) j ∈ Z ∈ S ( Π ) defines a unique billiardtrajectory γ as shown in Theorem 3.3. Therefore with corresponding ...a − .a a ... , we get a uniquebase symbol a , which further picks a base arc ( θ, φ ) on γ , thereby giving us a unique pointedgeodesic in G i.e. h ( γ, ( θ, φ )) = ...a − .a a ...h ○ τ (( γ, ( θ, φ ))) = h ( τ (( γ, ( θ, φ ))))= h (( γ, T ( θ, φ )))= h (( γ, ( θ , φ )))= ...a T − ( θ ,φ ) .a ( θ ,φ ) a T ( θ ,φ ) ... = ...a T − T ( θ,φ ) .a T ( θ,φ ) a T T ( θ,φ ) ... = ...a ( θ,φ ) .a T ( θ,φ ) a T ( θ,φ ) ... = σ ( h ( γ, ( θ, φ )))= σ ○ h ( γ, ( θ, φ )) Therefore, h ○ τ = σ ○ h , implying that h is a homomorphism.Consider an open set V = B (cid:15) ( γ, ( θ, φ )) in G . Thus ( γ ′ , ( θ ′ , φ ′ )) ∈ V iff d ∂ D ( θ, θ ′ ) , d ∂ D ( φ, φ ′ ) < (cid:15). Since the vertices are rational, it allows us to tesselate D with Π and its copies generated byreflecting Π about its sides and doing the same for the reflected copies along the unfolded geodesicgenerated by γ . Let us label the vertices of Π in anticlockwise sense by A , A , ...., A k and thevertices of i th copy of Π by A i , A i , ...., A ik . We note here that as we replicate Π along the unfolded27eodesic γ , it shrinks to a point both in future and past in euclidean sense. Define p to be thelargest positive integer such that A i , A i , ...., A ik /∈ ( θ − (cid:15), θ + (cid:15) ) ⨉( φ − (cid:15), φ + (cid:15) ) ∀ i = − p, − p + , ..., , , ..., p. Then h − ([ x − p ...x − x ...x k ]) ⊆ V. Therefore, h − is continuous.Figure 5: (cid:15) -tube about an unfolded billiard trajectoryConsider an open set U = [ x − m ...x − .x ...x m ] in ( X, σ ) . Pick an arbitrary bi-infinite sequence x ∈ U. Then from corresponding ( x n ) n ∈ Z , we geta billiard trajectory γ using Theorem 3.3. By pointing out the base arc ( θ, φ ) corresponding tosymbol x , we get a pointed geodesic ( γ, ( θ, φ )) that corresponds to a bi-infinite sequence whichwe label as y = ( y i ) . Now, in general y may not be in U , but since x and y belong to sameequivalence class, ∃ an s such that y [ s − m,s + m ] equals x − m ...x − .x ...x m . Therefore, ( γ, T − s ( θ, φ )) has its associated billiard bi-infinite sequence h ( γ, T − s ( θ, φ )) ∈ U. We construct m future and m past copies of Π in D by reflecting Π about its sides as suggested by h ( γ, T − s ( θ, φ )) ∈ U . Label T − s ( θ, φ ) as ( θ ′ , φ ′ ) .Let δ be defined as follows : δ = min i ∈{ ,...,k } { d ∂ D ( A mi , φ ′ ) , d ∂ D ( A − mi , θ ′ )} . Choose (cid:15) such that 0 < (cid:15) < δ . If ( γ ′ , ( θ ′ , φ ′ )) ∈ B (cid:15) ( γ, ( θ, φ )) , then [ h ( γ ′ , ( θ ′ , φ ′ ))] [− m,m ] = x − m ...x − x ...x m . h ( γ ′ , ( θ ′ , φ ′ )) ∈ U, i.e. h ( B (cid:15) ( γ, ( θ, φ ))) ⊆ U. Therefore, h is continuous.Thus, ( G , τ ) ≃ ( X, σ ) . Remark 3.1.
We note that both S ( Π ) and X generate the same language. S ( Π ) gives the trajec-tories whereas X gives each point in the trajectory. For a compact polygon with k vertices, the space of pointed bi-infinite sequences has forbidden set F = { , , ..., kk, ⋯·„„„„„„‚„„„„„„¶ ( + λ )− times , ⋯·„„„„„„‚„„„„„„¶ ( + λ )− times , ⋯·„„„„„„‚„„„„„„¶ ( + λ )− times , ⋯·„„„„„„‚„„„„„„¶ ( + λ )− times , ..., k k ⋯·„„„„„„„‚„„„„„„„¶ ( + λ k )− times , k k ⋯·„„„„„„„‚„„„„„„„¶ ( + λ k )− times } . Example 3.1.
We consider a particular case here where we consider a compact triangle with λ = λ = λ = . Here, we can declare X to be a − step SF T with F + = { wiiw ′ † − times ∀ i, ∀ w, ∀ w ′ , , , , , , } . Since X is a − step SF T , we go for X [ + ] = X G .Note that the associated transition matrix A here is aperiodic, and so ( X, σ ) is mixing. Alsothis A has a positive Perron eigenvalue which gives a positive topological entropy for ( X, σ ) .Thus here ( G , τ ) is a mixing system with positive topological entropy. In general both ( G , τ ) and ( X, σ ) will be mixing systems, that has been proved later in Theorem3.7. And so will have positive topological entropy. After establishing the codes for the billiards on ideal and compact rational polygons, we now do thesame for the case where some vertices of the polygon Π sit on ∂ D and some in D such that thesepolygons also tile D . We call such polygons semi-ideal rational . The coding rules for the billiardson such polygons is the natural amalgamation of the rules from the ideal and the compact rationalpolygon case. Theorem 3.5.
Let Π ⊂ D be a semi-ideal rational polygon with anti-clockwise enumeration of sideslabeled , , ..., k . Label the vertices of Π as v ,..., v k with Ω ,..., Ω k being the respective interior anglessuch that the adjacent sides of v i are i and i + . Further, assume that v i ∈ D ∀ i ∈ Λ ⊂ { , ..., k } and v i ∈ ∂ D ∀ i ∈ { , ..., k } − Λ with λ i = π / Ω i ∈ N ∀ i ∈ Λ . Then an equivalence class [ ...a − .a a ... ] with ...a − .a a ... ∈ { , ..., k } Z is in S ( Π ) (the space of all equivalence classes of pointed billiard bi-infinitesequences of Π ) if and only if ...a − .a a ... does not contain any immediate repetitions of symbols i.e., a j ≠ a j + ∀ j ∈ Z .(2) ...a − .a a ... does not contain more than λ i repetitions of two successive symbols i and i + forevery i ∈ Λ .(3) ...a − .a a ... does not contain an infinitely repeated sequence or bi-sequence of labels of twoadjacent sides i and i + ∀ i ∈ { , ..., k } − Λ .Moreover, every equivalence class of such billiard induced bi-infinite sequences corresponds to oneand only one billiard trajectory.Proof. Working on the similar lines, as for the case of ideal and compact case, we will start withestablishing the necessity of (1), (2) and (3). Consider a bi-infinite sequence ...a − .a a ... attachedto a billiard trajectory. Since, two geodesics in a hyperbolic plane can have at most one intersection,therefore (1) holds. Now, suppose (2) does not hold for ...a − .a a ... with [ ...a − .a a ... ] ∈ S ( Π ) . This means that for some i ∈ Λ, there exists a subword w of ...a − .a a ... with µ i (> λ i ) repetitionsof letters i and i +
1. Following the same line of argument as in Theorem 3.3, we get a contradiction.Suppose (3) does not hold for ...a − .a a ... with [ ...a − .a a ... ] ∈ S ( Π ) . This means that for some i ∈ { , ...k } − Λ, it contains an infinitely repeated sequence or bi-sequenceof labels of sides i and i +
1. Under Cayley transformation, we can shift the whole billiard tableto H and an appropriate isometry of H allows us to assume that the two sides i and i + H by the vertical lines x = x =
1. Now, our trajectory hits these two sidesrepeatedly infinite number of times. if we unfold this part of the trajectory in H , the unfolded partof the trajectory being a geodesic lies on a semi-circle centered at real axis. Since a semi-circle canintersect only finitely many lines from the family { x = n ∶ n ∈ N } , therefore we get a contradictionto our assumption that the trajectory has infinitely many parts consisting of consecutive hits onsides i and i +
1. Thus, the necessity of conditions (1), (2) and (3) is established. Now suppose ...a − .a a ... be a pointed bi-sequence for which (1), (2) and (3) hold. We will construct a uniquebilliard trajectory defined by it in Π. We structure the rest of the proof on the similar lines as in3.3, going through the three main stages. First, we will prove that if we consider a bi-sequencesatisfying (1), (2) and (3) which is periodic, it uniquely defines a billiard trajectory. Then, wewill show that the set of periodic bi-infinite sequences satisfying (1), (2) and (3) is dense in theset of all bi-infinite sequences satisfying (1), (2) and (3). Lastly, this will allow us to construct aunique billiard trajectory against an arbitrary bi-sequence satisfying (1), (2) and (3) as the limitof a sequence comprising of periodic bi-infinite sequences satisfying (1), (2) and (3).The set of all periodic bi-infinite sequences is dense in a full k-shift. If we remove bi-infinitesequences containing words ( i i + ) µ i and ( i + i ) µ i where µ i > λ i = π / Ω i for some i ∈ Λ, orcontaining an infinitely repeated sequence or bi-sequence of labels of two adjacent sides i and i + i ∈ { , ..., k } − Λ, or the ones for which a i = a i + for any i , the set of all periodic bi-infinitesequences in the remaining space is still dense. Indeed, if we start with ...a − .a a ... satisfying (1),(2) and (3), we can define a sequence of periodic points as follows: take x = ( a − .a a ) ∞ , x = ( a − a − .a a a ) ∞ ...a − .a a ... itself satisfies (2) and(3) which comprises of periodic points and converges to ...a − .a a ... . Thus, we have the requireddenseness property.Suppose ...a − .a a ... is an arbitrary bi-sequence obeying (1), (2) and (3). Let ( x m ) m ∈ N bea sequence generated by the above construction, then x m → ...a − .a a ... . Now, each x m hasa unique geodesic γ m associated with it. Since, D is geodesically complete, the sequence ofgeodesics, ( γ m ) m ∈ N converges to a limit geodesic γ , which acts as the unique geodesic associatedwith ...a − .a a ... . On folding γ back into the fundamental polygon Π, we get a unique billiardtrajectory associated with [ ...a − .a a ... ] . Theorem 3.6.
Let Π ⊂ D be a semi-ideal rational polygon with anti-clockwise enumeration of sideslabeled , , ..., k . Label the vertices of Π as v ,..., v k with Ω ,..., Ω k being the respective interior anglessuch that the adjacent sides of v i are i and i + . Further, assume that v i ∈ D ∀ i ∈ Λ ⊂ { , ..., k } and v i ∈ ∂ D ∀ i ∈ { , ..., k } − Λ with λ i = π / Ω i ∈ N ∀ i ∈ Λ . Let G be the space of pointed geodesicson Π and X the corresponding space of all bi-infinite sequences ...a − .a a ... ∈ { , ..., k } Z satisfying(1), (2) and (3) from Theorem 3.5. Then ( G , τ ) ≃ ( X, σ ) .Proof. Define h ∶ ( G , τ ) → ( X, σ ) by h ( γ, ( θ, φ )) = ...a T − ( θ,φ ) .a ( θ,φ ) a T ( θ,φ ) ...h ( γ, ( θ, φ )) = h ( γ ′ , ( θ ′ , φ ′ ))⇒ ...a T − ( θ,φ ) .a ( θ,φ ) a T ( θ,φ ) ... = ...a T − ( θ ′ ,φ ′ ) .a ( θ ′ ,φ ′ ) a T ( θ ′ ,φ ′ ) ... ⇒ ( a T n ( θ,φ ) ) n ∈ Z = ( a T n ( θ ′ ,φ ′ ) ) n ∈ Z From Theorem 3.5 , we see that ( T n ( θ, φ )) n ∈ Z = ( T n ( θ ′ , φ ′ )) n ∈ Z and a ( θ,φ ) = a ( θ ′ .φ ′ ) ⇒ ( γ, ( θ, φ )) = ( γ ′ , ( θ ′ , φ ′ )) . This gives the injectivity of h.The surjectivity of h is established from the fact that each ( a j ) j ∈ Z ∈ S ( Π ) defines a unique billiardtrajectory γ as shown in Theorem 3.5. Therefore with corresponding ...a − .a a ... , we get a uniquebase symbol a , which further picks a base arc ( θ, φ ) on γ , thereby giving us a unique pointedgeodesic in G i.e. h ( γ, ( θ, φ )) = ...a − .a a ... ○ τ (( γ, ( θ, φ ))) = h ( τ (( γ, ( θ, φ ))))= h (( γ, T ( θ, φ )))= h (( γ, ( θ , φ )))= ...a T − ( θ ,φ ) .a ( θ ,φ ) a T ( θ ,φ ) ... = ...a T − T ( θ,φ ) .a T ( θ,φ ) a T T ( θ,φ ) ... = ...a ( θ,φ ) .a T ( θ,φ ) a T ( θ,φ ) ... = σ ( h ( γ, ( θ, φ )))= σ ○ h ( γ, ( θ, φ )) Therefore, h ○ τ = σ ○ h , implying that h is a homomorphism.Consider an open set V = B (cid:15) ( γ, ( θ, φ )) in G . Thus ( γ ′ , ( θ ′ , φ ′ )) ∈ V if and only if d ∂ D ( θ, θ ′ ) , d ∂ D ( φ, φ ′ ) < (cid:15). Since the vertices are rational or ideal, it allows us to tesselate D with Π and its copies generated byreflecting Π about its sides and doing the same for the reflected copies along the unfolded geodesicgenerated by γ . Let us label the vertices of Π in anticlockwise sense by A , A , ...., A k and thevertices of i th copy of Π by A i , A i , ...., A ik . We note here that as we replicate Π along the unfoldedgeodesic γ , it shrinks to a point both in future and past in euclidean sense. The vertices that lieon ∂ D stay put, whereas the ones lying in D move towards the boundary under both future andpast limits. Define p to be the largest positive integer such that A i , A i , ...., A ik /∈ ( θ − (cid:15), θ + (cid:15) ) ⨉( φ − (cid:15), φ + (cid:15) ) ∀ i = − p, − p + , ..., , , ..., p. Then h − ([ x − p ...x − x ...x k ]) ⊆ V. Therefore, h − is continuous.Consider an open set U = [ x − m ...x − .x ...x m ] in ( X, σ ) . Pick an arbitrary bi-infinite sequence x ∈ U. Then from corresponding ( x n ) n ∈ Z , we geta billiard trajectory γ using Theorem 3.5. By pointing out the base arc ( θ, φ ) corresponding tosymbol x , we get a pointed geodesic ( γ, ( θ, φ )) corresponding to the bi-infinite sequence whichwe label as y = ( y i ) . Now, in general y may not be in U , but since x and y belong to sameequivalence class, ∃ an s such that y [ s − m,s + m ] equals x − m ...x − .x ...x m . Therefore, ( γ, T − s ( θ, φ )) has its associated pointed billiard bi-sequence h ( γ, T − s ( θ, φ )) ∈ U . We construct m future and mpast copies of Π in D by reflecting Π about its sides as suggested by h ( γ, T − s ( θ, φ )) ∈ U. (cid:15) -tube about an unfolded billiard trajectoryLabel T − s ( θ, φ ) as ( θ ′ , φ ′ ) . Let δ be defined as follows : δ = min i ∈{ ,...,k } { d ∂ D ( A mi , φ ′ ) , d ∂ D ( A − mi , θ ′ )} . Choose (cid:15) such that 0 < (cid:15) < δ . If ( γ ′ , ( θ ′ , φ ′ )) ∈ B (cid:15) ( γ, ( θ, φ )) , then [ h ( γ ′ , ( θ ′ , φ ′ ))] [− m,m ] = x − m ...x − x ...x m . Thus h ( γ ′ , ( θ ′ , φ ′ )) ∈ U, i.e. h ( B (cid:15) ( γ, ( θ, φ ))) ⊆ U. Therefore, h is continuous.Thus ( G , τ ) ≃ ( X, σ ) .We note that if ˜ X is the closure of X in { , , . . . , k } Z , then ( ˜ X, σ ) is an SFT. We note that in case of rational polygons the SFT ( ˜ X, σ ) is mixing. We present a proof for thesemi-ideal case and note that it works the same for the compact case(where ˜ X = X ).33 heorem 3.7. Let Π ⊂ D be a semi-ideal rational polygon with anti-clockwise enumeration of sideslabeled , , ..., k . Label the vertices of Π as v ,..., v k with Ω ,..., Ω k being the respective interior anglessuch that the adjacent sides of v i are i and i + . Further, assume that v i ∈ D ∀ i ∈ Λ ⊂ { , ..., k } and v i ∈ ∂ D ∀ i ∈ { , ..., k } − Λ with λ i = π / Ω i ∈ N ∀ i ∈ Λ . Let G be the space of pointed geodesicson Π and X the corresponding space of codes ...a − .a a ... ∈ { , ..., k } Z satisfying (1), (2) and (3)from 3.5. Then ( ˜ X, σ ) is mixing.Proof. We start with assumption that k ≥
6. For every pair of words u, v ∈ L( ˜ X ) , our aim is todeclare an N ∈ N such that a w ∈ L( ˜ X ) can be chosen with ∣ w ∣ = n for all n ≥ N with uwv ∈ L( ˜ X ) .Suppose the word u ends in ri and the word v starts with jl , then we can choose two distictlabels p, q distinct from r, i, j, l . We construct w as ( pq ) d i ( pq ) d i... ( pq ) d , where the ( pq ) blocks arerepeated c times. Here, c can be chosen arbitrarily large but d has a natural restriction for adjacent p, q under the coding rules that we have laid down in Theorem ?? . This establishes the mixingproperty for any ˜ X associated with a semi-ideal rational polygon with sides equal or more than 6.Next, we consider the case where k =
5. Under the same notation as above, suppose all of r, i, j, l are distinct. Then, we construct w as ( pj ) d i ( pj ) d i... ( pj ) d i by choosing p distinct from r, i, j, l . If two of r, i, j, l are equal, say i = l ≠ j, r , then we choose p, q ≠ i, l, r and construct w as ( pq ) d i ( pq ) d i... ( pq ) d . Similar constructions can be done for w is any other similarities occur among r, i, j, l as we get more room to wiggle around. This establishes the mixing property for any ˜ X associated with a semi-ideal rational polygon with 5 sides.Now, we consider the case k =
4. The situations where more than one similarities occuramong r, i, j, l can be handled in same way as discussed above. If all of r, i, j, l are distinct, thenwe construct w as ( jl ) d i ( jl ) d i... ( jl ) d i ( ri ) e where d, e are chosen respecting the respective cod-ing rules. If exactly two of r, i, j, l are equal, say i = l , then we choose p ≠ i, j, r and construct w = ( pr ) d i ( pr ) d i... ( pr ) d i ( pr ) d . Other cases of similarity among r, i, j, l can be handled in the sameway. This establishes the mixing property for any ˜ X associated with a semi-ideal rational polygonwith 4 sides.Lastly, for k = r, i, j, l are same. If exactly two are same, say i = j , thenwe construct w as ( lr ) d i ( lr ) d i... ( lr ) d . Similarly, if i = l , then we construct w as ( jr ) d i ( jr ) d i...i ( jr ) d .If two matchings occur among r, i, j, l , say j = r, l = i , then we choose p ≠ i, r and construct w as ( pr ) d i ( pr ) d ...i ( pr ) d i ( rp ) d . If more than two matchings occur among r, i, j, l , it allows us to choosedistinct p, q from r, i, j, l as we get more room to wiggle around and similar constructions for w followas discussed above. This establishes the mixing property for any ˜ X associated with a semi-idealrational polygon with 3 sides. Remark 3.2.
Note that this also gives mixing for ( X, σ ) in the compact rational case. In case ofsemi-ideal rational polygons, the set ˜ X ∖ X consists of strings of the form i ( i + ) as described inTheorem 3.5. This implies that ( X, σ ) is also mixing.Thus ( G , τ ) is mixing. This also gives positive topological entropy for ( G , τ ) .34 Convergence in the Space of all Subshifts and Hyperbolic Poly-gons
Let S be the set of all shift spaces of bi-infinite sequences defined over a finite alphabet withcardinality atleast 2.For k ≥
2, let A k = { , , . . . , k } . Define X k = A Z k , for each k ≥
2, the space of all bi-infinitesequences taking values in { , , . . . , k } indexed by Z , endowed with the product topology. Then,for each k ≥ X k is metrizable with the metric defined in Equation (1), and can be taken as acompact metric space. Thus for each k ≥
2, we have subshifts ( X k , σ ) .Now each X k ⊊ N Z , k ≥
2. Any shift space Y ∈ S can be considered isometric to a subshift X ⊆ X k for some k ≥
2. Thus with respect to the Gromov-Hausdorff metric as defined in Equation(4), every subshift in S is isometrically imbedded in N Z . Thus N Z can be considered as the universalUrysohn space for all subshifts of bi-infinite sequences over a finite alphabet. Every such shift space Y ∈ S is isometric to a compact subset of N Z . Hence we can consider S ⊊ K( N Z ) .Thus each subshift ( Y, σ ) , for Y ∈ S , can be considered as a subsystem of ( N Z , σ ) . We hencecall ( N Z , σ ) as the universal shift space , and every subshift from S is its subsystem.Consider the metric space (K( N Z ) , d H ) . Thus d H gives a metric on S and we lift that metricon the set of subshifts ( Y, σ ) for Y ∈ S . We note that this lift is in the sense of Gromov since weidentify isometric shift spaces in S with some subset of X k for some k .Our special emphasis will be on sft ⊊ S , where sft is the subclass of all subshifts of finite typein S . We see that sft is a very important subset of S . Lemma 4.1.
For every distinct
X, Y ∈ sft , d H ( X, Y ) > .Proof. Suppose that there exists no δ > d H ( X, Y ) > δ . Then for every x ∈ X , thereexists a y ∈ Y such that d ( x, y ) < δ . This means that ∃ k > x [− k,k ] = y [− k,k ] . Taking δ →
0, this means that both x, y have the same words of length 2 k +
1, for large k . This implies x = y , i.e. we get X = Y . Theorem 4.1. sft is a dense subset of S .Proof. Let X ∈ S ∖ sft be a shift space and let F be an infinite collection of blocks such that X = X F .We borrow this idea from [35]. The elements of F can be arranged lexicographically, andaccordingly let F = { f , f , . . . , f n , . . . } .Define F j = { f , . . . , f j } for j ≥
1. Then X = ⋂ j ∈ N X F j . Thus, X F j → X . 35e try to understand the space S ⊊ K( N Z ) .Suppose X n → X in S . Then there exists a sequence { k n } ↗ ∞ such that ∀ x ∈ X ∃ x n ∈ X n such that x [− k n ,k n ] = x n [− k n ,k n ] . We note that the set of finite subsets of N Z is dense in K( N Z ) . But all of them are not containedin S . We consider Z ∈ S such that ∣ Z ∣ < ∞ . Since σ ( Z ) ⊂ Z , and σ is a homeomorphism, σ justpermutes the elements of Z . Thus, Z is just a periodic orbit or a finite union of periodic orbits.Also, this Z can be considered as a vertex shift over a finite alphabet and so Z ∈ sft .Again we note that for every periodic orbit O( x ) ∈ N Z , there exists no other element of S thatcan be contained in [O( x )] (cid:15) for every (cid:15) >
0. Thus each finite element of sft is isolated.Let X ∈ sft be infinite. Then there is a finite F such that X = X F . Let A ⊂ N be the alphabetset for X . Then there exists a, b ∈ A , a ≠ b with ab N ∉ F for some N ∈ N . Else X will be finite.Define F j = F ∪ { ab N + j } . Then X F ⫋ X F ⫋ . . . ⫋ X F j ⫋ . . . and X = ⋃ j ∈ N X F j . Hence X F j → X .This gives the complete topological structure of the space S . Thus, we can say: Theorem 4.2.
Every finite element of S is isolated and every infinite element is an accumulationpoint in S .Further, for every infinite X ∈ S there exists a sequence { X F j } in sft such that X F j → X . Let us look into the statement “ X n → X in S ”. Definition 4.1.
We say that the subshift ( X, σ ) is the limit of subshifts {( X n , σ )} and write lim n →∞ ( X n , σ ) = ( X, σ ) in ( N Z , σ ) if and only if X n → X in S . Remark 4.1.
We note that ( X n , σ ) → ( X, σ ) ≅ X n → X . We compare the dynamical properties of the sequence {( X n , σ )} of subshifts in sft to those ofthe limit ( X, σ ) . We look into the dynamical properties of the limit ( X, σ ) that are inherited bythis limiting process. Remark 4.2.
We note that as mentioned in [35], if X n → X then the entropies h ( X n ) → h ( X ) .By the observation in the previous section, for every X ∈ S , we can choose a sequence { X n } in sft with X n → X . Now h ( X n ) = log λ n where λ n is the Perron eigenvalue corresponding to X n ∈ sft ,for every n . Thus, each h ( X ) can be considered as a limit of a sequence of the form log λ n .This also implies that for X ∈ S , if X n → X with lim inf n →∞ h ( X n ) > then h ( X ) > . Proposition 4.1. If X = O( x ) for some x ∈ N Z , then there exists a sequence { X n } in sft with ∣ X n ∣ < ∞ such that X n → X . roof. For some K > u = x [− K,K ] , u = x [− K − ,K + ] , . . . , u n = x [− K − n,K + n ] , . . . .Define X n = O( u ∞ n ) , for j ≥
1, where a ∞ = . . . aaaaaaa . . . the infinite concatenation of a withitself. Since each X n is finite, it is an SFT.Now X = O( x ) and so X n → X . Remark 4.3.
We note that when ( X, σ ) is minimal, for some X ∈ S , X is obtained as a limit ofa sequence of finite SFTs. Now if X has no isolated points then X is sensitive. Since a finite SFTis always equicontinuous, we note that equicontinuity is not preserved by taking limits. However,recurrence is preserved on taking limits. We can say more:
Theorem 4.3.
For X ∈ S , if X n → X with each ( X n , σ ) being a non-wandering SFT, then ( X, σ ) is chain recurrent.Proof. Since X n → X , for given (cid:15) >
0, there exists N > X n ⊂ X (cid:15) for n > N .Thus for x ∈ X , there exists x n ∈ X n such that for n > N , x n is in (cid:15) − neighbourhoods of x and x n → x .Thus there exist sequence { k n } ↗ ∞ such that x [− k n ,k n ] = x n [− k n ,k n ] . Since each X n is non-wandering, for x n ∈ X n there exists M n > z n ∈ X n such that z n [− k,k ] = x n [− k,k ] (cid:212)⇒ z n [ M n − k,M n + k ] = x n [− k,k ] Choose m > N such that k m > k , then there exists L , . . . , L t > z , . . . , z t ∈ X such that z [− k,k ] = x [− k,k ] , z [ L − k,L + k ] = z [− k,k ] , . . . , and z t − [ L t − − k,L t − + k ] = z t [− k,k ] with z t [ L t − k,L t + k ] = x [− k,k ] . This proves the chain recurrence of X . Theorem 4.4.
For X ∈ S , if X n → X with each ( X n , σ ) a transitive(irreducible) SFT, then ( X, σ ) is chain transitive.Proof. Since X n → X , for given (cid:15) >
0, there exists N > X n ⊂ X (cid:15) for n > N .Thus for x, y ∈ X , there exists x n , y n ∈ X n such that for n > N , x n , y n are in (cid:15) − neighbourhoodsof x, y and x n → x and y n → y .Thus there exist sequence { k n } ↗ ∞ such that x [− k n ,k n ] = x n [− k n ,k n ] & y [− k n ,k n ] = y n [− k n ,k n ] . Since each X n is transitive, for every x n , y n ∈ X n there exists M n > z n ∈ X n such that z n [− k,k ] = x n [− k,k ] (cid:212)⇒ z n [ M n − k,M n + k ] = y n [− k,k ] Choose m > N such that k m > k , then there exists L , . . . , L t > z , . . . , z t ∈ X such that z [− k,k ] = x [− k,k ] , z [ L − k,L + k ] = z [− k,k ] , . . . , and z t − [ L t − − k,L t − + k ] = z t [− k,k ] with z t [ L t − k,L t + k ] = y [− k,k ] . This proves the chain transitivity of X . 37nd the discussions in Remark 2.1 gives, Corollary 4.1. If X n → X with each ( X n , σ ) a transitive(irreducible) SFT and X ∈ sft , then ( X, σ ) is transitive. Example 4.1.
We note that the limit of transitive SFTs need not be transitive.We consider SFTs ( X n , σ ) ∀ n ∈ N , with alphabet { , , } and forbidden words F n = { k , k , , , , , k , k ∶ k ≤ n } respectively.We note that in each X n , a or must be followed by a block of at least n + s .Also X ⫋ X ⫋ . . . and X n → X where X = ⋂ n ∈ N X n = { ∞ } ∪ { σ n ( ∞ . ∞ ) ∶ n ∈ Z } ∪ { σ n ( ∞ . ∞ ) ∶ n ∈ Z } . Each ( X n , σ ) is transitive but ( X, σ ) is not transitive.We note that ( X, σ ) is chain transitive. Example 4.2.
We note that the limit of transitive SFTs can be transitive even when not SFT.Let ( X, σ ) be the even shift with alphabet { , } and forbidden words F = { k ∶ k ∈ N + } .We consider SFTs ( X n , σ ) ∀ n ∈ N , with alphabet { , } and forbidden words F n = { k ∶ k ∈ N + and k ≤ n } respectively.We note that in each X n , a must be followed by a block of at least n + s .Also X ⫌ X ⫌ . . . and X n → X .We note that each ( X n , σ ) is transitive and ( X, σ ) is also transitive. Theorem 4.5.
For X ∈ S , if X n → X with each ( X n , σ ) a mixing SFT, then ( X, σ ) is chainmixing.Proof. Since X n → X , for given (cid:15) >
0, there exists N > X n ⊂ X (cid:15) for n > N .Thus for x, y ∈ X , there exists x n , y n ∈ X n such that for n > N , x n , y n are in (cid:15) − neighbourhoodsof x, y and x n → x and y n → y .Thus there exist sequence { k n } ↗ ∞ such that x [− k n ,k n ] = x n [− k n ,k n ] & y [− k n ,k n ] = y n [− k n ,k n ] . Since each X n is mixing, for every x n , y n ∈ X n there exists M n > z n t ∈ X n for all N t ≥ M n such that z n t [− k,k ] = x n [− k,k ] (cid:212)⇒ z n t [ N t − k,N t + k ] = y n [− k,k ] Choose m > N such that k m > k , then there exists M >
0, and w , . . . , w j ∈ X and N , . . . , N j > j ≥ M , such that w [− k,k ] = x [− k,k ] w [− k,k ] = w [ N − k,N + k ] . . .w j [ N j − k,N j + k ] = y [− k,k ] . X .And the discussions in Remark 2.1 gives, Corollary 4.2. If X n → X with each ( X n , σ ) a mixing SFT and X ∈ sft , then ( X, σ ) is mixing. Remark 4.4.
We note that Example 4.1 gives an example of mixing SFTs whose limit is chainmixing but not mixing. Example 4.2 gives an example of mixing SFTs whose limit is mixing thoughnot SFT. Also Example 4.1 and Example 4.2 both illustrate Theorem 2.2.
We can say something more here. Let X n → X and let x, y ∈ X . For (cid:15) > B (cid:15) ( x ) , B (cid:15) ( y ) denote the (cid:15) -balls centered around x, y respectively in N Z .If ( X, σ ) is also transitive then we note that the hitting times N ( B (cid:15) ( x ) ∩ X, B (cid:15) ( y ) ∩ X ) = lim n →∞ N ( B (cid:15) ( x ) ∩ X n , B (cid:15) ( y ) ∩ X n ) where this limit is taken in 2 N .This is simple to observe since for every (cid:15) >
0, and x, y ∈ X , there exists an N > B (cid:15) ( x ) ∩ X n ≠ ∅ , and B (cid:15) ( y ) ∩ X n ≠ ∅ for n ≥ N . The results that we have established in Theorems 3.2, 3.4, 3.6 indicate the coming together ofgeometric and dynamical convergence for the class of semi-ideal rational polygons in a sense thatwe discuss ahead. We construct a sequence of polygons Π n in the semi-ideal rational class thatconverges to a polygon Π in the same class. Then, our results indicate that the correspondingsequence of compactifications of the space of codes ˜ X n must converge to the compactification ofthe space of codes for Π.For a sequence of polygons { Π n } n ∈ N and a polygon Π with k vertices in D , we label the verticesof each Π n by v ni with i ∈ { , , ..., k } and the vertices of Π by v i with i ∈ { , , ..., k } . If v ni → v i for each i , then we call it the convergence of a polygonal sequence and denote it as Π n → Π. Thisconvergence is in accordance with the
Gromov-Hausdorff metric as given in Equation 4.We consider the full shift on k symbols and denote the space of all compact subsets of { , , ..., k } Z by M . In view of the above discussion, we talk about the convergence of a sequence of compact-ifications of the spaces of codes of the corresponding polygonal billiard tables under the ambit ofthe metric structure provided by (M , d GH ) . It is to be noted here that for a polygonal billiardtable of above mentioned class, the corresponding compactification of the space of codes uniquelydetermines the space of codes itself. This will allow us to trace back our sequence of polygonsstarting with the convergence of a sequence of associated compactifications of the space of codes ina sense that we describe ahead. Theorem 4.6.
Let { Π n } n ∈ N be a sequence of semi-ideal rational polygons in D each having k vertices and Π be another such polygon. Further, suppose X n be the corresponding space of codesfor Π n and ˜ X n the corresponding compactification, for each n ∈ N . Let X be the space of codes for Π with compactification ˜ X . Then Π n → Π implies ˜ X n → ˜ X . roof. Let Π n → Π then v ni → v i ∀ i ∈ { , , ..., k } . We note here that for the given counter-clockwiselabelling of each Π n , we label Π as suggested by the above convergence. Consider any arbitrary j ∈ { , , ..., k } . If Ω j is an ideal vertex, then with Ω nj →
0, we get ∀ (cid:15) > , ∣ Ω nj − ∣ < (cid:15) for sufficientlylarge n . Now, moving to a subsequence with non-zero terms, if needed, we get λ nj = π / Ω nj > / (cid:15) forsufficiently large n . Note that the zero terms of the sequence, if present satisfy the below mentionedcriteria vacuously. Thus, for sufficiently large n , we have ( j, j + ) λ nj + ∈ (L( X n )) c . Therefore, as n → ∞ , (L( X n )) c contains ( j, j + ) p ∀ p ∈ N .If Ω j is a non-zero rational, then ∀ (cid:15) > , ∣ Ω nj − Ω j ∣ < (cid:15) for sufficiently large n . Say, Ω j = π / m ,then if (cid:15) < π / m ( m + ) , we get Ω nj = Ω j for sufficiently large n . Thus, λ nj = λ j for sufficiently large n , which further implies that ( j, j + ) λ j ∈ (L( X n )) c for sufficiently large n . Therefore, ˜ X n → ˜ X in d GH metric. Remark 4.5.
We note that we can recover X n and X from ˜ X n and ˜ X to see that X n → X in thesense of Gromov. Remark 4.6.
The converse of the Theorem 4.6 is more subtle. We describe the case of an idealtriangle Π . We can assume that Π is so placed that it contains the center of D . This can be doneas the coding rules are not dependent on the position of the vertices, thereby the polygon can betweaked as required. Its corresponding ˜ X is described by the forbidden set F ˜ X = { , , } . Letus pick up a sequence ˜ X n given by the corresponding forbidden sets F ˜ X n = { , , , ( ) n + } .Then ˜ X n → ˜ X in the d GH metric. For ˜ X the corresponding X is determined by the coding rulesof 3.2. We call the center of D as O and construct the radial Euclidean lines Ov , Ov , Ov . Next,we choose a sequence of polygons Π n by taking v n ≡ v , v n ≡ v and placing v n on Ov . Note thatfor each n the choice of v n is unique under the requirement that Ω n = π /( n + ) . This ensures that Π n → Π and gives us a partial answer to the converse of Theorem 4.6. The same can be said aboutan arbitrary ideal polygon. We also remark here that the above construction still holds good for anysemi-ideal rational polygon with the restriction that exactly one vertex is in D with all other verticessitting on ∂ D . Acknowledgements
We thank
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