Bridge to Hyperbolic Polygonal Billiards
BBRIDGE TO HYPERBOLIC POLYGONAL BILLIARDS
HASSAN ATTARCHI AND LEONID A. BUNIMOVICH
Abstract.
It is well-known that billiards in polygons cannot be chaotic(hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonalbilliard is zero. We consider physical polygonal billiards where a movingparticle is a hard disc rather than a point (mathematical) particle andshow that typical physical polygonal billiard is hyperbolic at least ona subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle (providedthat the particle is not so big that it cannot move within a polygon).This happens because a typical physical polygonal billiard is equivalentto a mathematical (point particle) semi-dispersing billiard. We alsoconjecture that in fact typical physical billiard in polygon is ergodicunder the same conditions. Introduction
In the last years, there has been significant research on billiards in poly-gons [3, 6, 15, 16, 18, 24, 25]. Dynamics of these models is extremely difficultto rigorously analyze which often happens with systems with intermediate,neither regular (integrable) nor chaotic, behavior. Billiards were introducedand successfully used as models for numerous phenomena and processes innature, especially in physics [14, 17, 20, 22, 26]. However, only mathemati-cal billiards were considered, where a point (mathematical) particle moves.There are no and there will be no such particles in reality. Nevertheless,studies of just mathematical billiards were considered to be sufficient. Thereasons for that were twofold. First, a system with real (finite size) particlescould be sometimes reduced to a mathematical (point particle) billiard insome peculiar billiard table. Such situation takes place for instance for cel-ebrated Boltzmann gas of hard spheres [5]. Moreover, all basic examples ofbilliards with regular dynamics (e.g. billiards in circles and rectangles) haveabsolutely the same dynamics if one considers a (not too large) hard discmoving within the same billiard table. The same happens to the most pop-ular chaotic (hyperbolic) billiards like Sinai billiards and squashes (stadiumis a special case of a squash) [10, 23].It has been shown, however, that the transition to physical billiards cancompletely change the dynamics. Moreover, any type of chaos-order or
Mathematics Subject Classification.
Key words and phrases.
Mathematical billiards, Billiards in polygons, Physical bil-liards, Hyperbolicity, Kolmogorov-Sinai entropy.To Yakov Grigorievich Sinai on his 85 birthday. a r X i v : . [ m a t h . D S ] A ug H. ATTARCHI AND L. A. BUNIMOVICH order-chaos transition may occur [10]. In particular, it has been shown thatclassical Ehrenfests’ Wind-Tree gas has richer dynamics than the Lorentzgas if a moving particle is real (physical) [1]. In the present paper, we showthat typical physical billiard in polygons is chaotic for an arbitrarily smallsize (radius) of a moving particle. The last means that physical billiardsin generic polygons are hyperbolic on a subset of positive measure and,particularly, have a positive Kolmogorov-Sinai entropy to the contrary tomathematical billiards in polygons, which have zero KS-entropy.2.
Billiards in Polygons
Let P be the space of all closed polygons in R and P n ⊂ P denote thespace of all polygons with n vertices. Let { v , v , . . . , v n − } be the setof vertices of a polygon in P n . If we fix one side of this polygon on the x -axis and one of the vertices of that side at the origin (e.g. v = (0 ,
0) and v n − = ( x, P n → R n − induces a topology in P n such that its corresponding metric makes the space P n complete [27]. If allangles of a polygon are commensurate with π , then it is called a rationalpolygon. It is well-known that rational polygons are dense in P .A billiard in a polygon P ∈ P is a dynamical system generated by themotion of a point particle along a straight line inside P with unit speed andelastic reflections off its boundary ∂P .The phase space of this dynamical system isΛ P = { ( x, ϕ ) ∈ ∂P × ( − π , π x is not a vertex of P } , where ϕ is the reflection angle with respect to the inward normal vector n ( x )to the boundary at reflection point x ∈ ∂P .Let γ be a billiard orbit in the polygon P . If the orbit γ hits a side ofthe boundary ∂P then instead of reflecting the orbit γ off that side of P ,one may reflect P about that side. Denote the reflected polygon by P . Theunfolded orbit γ is a straight line as the continuation of γ in P . In thegeometric optics this procedure is called the method of images or unfolding[19]. Continuing this procedure for n consecutive reflections of the orbit γ ,we obtain a sequence of polygons P, P , P , . . . , P n where the unfoldedorbit γ is a straight segment through P to P n (Fig. 1).The unfolding process can also be done backward in time. A trajectorystops when it hits a vertex. The unfolded orbit γ is a finite segment if it hitsvertices of P both in the future and in the past. Such trajectories are called generalized diagonals . In [19, 27], it is shown that the set of generalizeddiagonals of the polygon P ∈ P is countable.Consider ( x, ϕ ) ∈ Λ P . A direction ϕ at point x is called an exceptionaldirection if its trajectory hits a vertex of P . It is not difficult to see thatthe number of these exceptional directions is countable at each point x . RIDGE TO HYPERBOLIC POLYGONAL BILLIARDS 3
P P P P P P g > Figure 1.
The unfolding process.3.
Physical Billiards in non-Convex Polygons
A physical billiard in a domain (billiard table) is generated by the motionof a hard ball (disc) of radius r > r >
0, it is enough to follow the motion of itscenter. We can see that the center of particle moves within a smaller billiardtable, which one gets by moving any point x of the boundary by r to theinterior of the billiard table along the internal normal vector n ( x ) [10].It is easy to see that dynamics of a physical billiard in a convex simplyconnected polygon is completely equivalent to dynamics of a mathematicalbilliard in this polygon [10]. However, the situation is totally different fornon-convex polygons (more precisely, polygons with at least one reflex angle,see Fig. 2). In this case, the boundary of the equivalent mathematicalbilliard acquires some dispersing parts, which are arcs of a circle of radius r (see e.g. [12, 13]). Figure 2.
To have dispersing parts in the boundary ofmathematical billiards equivalent to physical billiards in non-convex polygons, the particle has to be small enough and thepolygon has to have at least one reflex angle.Let P ref be the set of all polygons that they have at least one reflexangle. To show that P ref is dense in P , we use the metric d ( ., . ) on P whichis defined as:(3.1) d ( P, Q ) = (cid:90) R | χ P ( x ) − χ Q ( x ) | dx, H. ATTARCHI AND L. A. BUNIMOVICH where
P, Q ∈ P and, χ P ( x ) = (cid:26) x ∈ P, otherwise. The topology induced by the metric d ( ., . ) in P n is equivalent to the inducedtopology in P n by the embedding P n → R n − . Thus, rational polygonsare dense in P with respect to the metric d ( ., . ). Lemma 3.1. P ref is dense in P .Proof. Let P ∈ P be a polygon with n vertices { v , v , . . . , v n − } . Withoutloss of generality, we assume that the randomly chosen edge of P is v v n − .On the perpendicular bisector of v v n − in the interior of P , we choose asequence of points { v n k } ∞ k = k such that the reflex angle ∠ v v n k v n − = π + πk ( k is big enough to have v n k for k ≥ k in the interior of P , see Fig. 3). v v v n-1 n k Figure 3.
Replacing one edge of a polygon by a reflex angle.If we denote the non-convex polygons with vertices { v , v , . . . , v n − , v n k } by P k then it follows that P k ∈ P ref and d ( P, P k ) → , as k → ∞ . (cid:3) Lemma 3.2. P ref is open in P .Proof. It is easy to see that any perturbation of P ∈ P ref will have at leastone reflex angle. This means P ref is open in P . (cid:3) Continued fractions for billiards were introduced in Sinai’s fundamentalpaper [23]. They serve as a basic tool for analysis of billiards dynamics.Let 0 = t < t < t < . . . be the reflection times of the trajectory γ offthe boundary ∂Q where Q is an arbitrary billiard table. Denote by κ i thecurvature of the boundary at the i th reflection point with respect to theinward unit normal vectors n ( x ) to the boundary at x ∈ ∂Q , and by ϕ i the RIDGE TO HYPERBOLIC POLYGONAL BILLIARDS 5 i th reflection angle such that − π < ϕ i < π . The corresponding continuedfraction of this trajectory is given by κ = 1 τ + 12 κ cos ϕ + 1 τ + 12 κ cos ϕ + 1 τ + 1. . . , where τ i = t i − t i − for i = 1 , , . . . .Let P ∈ P ref , then the curvature of boundary components of the mathe-matical billiard equivalent to the physical billiard in P is either 0 or r . If anorbit hits the dispersing components infinitely many times where reflectionnumbers on dispersing parts are given by the sequence { i k } ∞ k =1 , then thecontinued fraction of this orbit will have the following form between i j thand i j +1 th reflections,. . . + 12 r cos ϕ i j + 1( τ i j +1 + · · · + τ i j +1 ) + 12 r cos ϕ i j +1 + 1. . . . All elements of continued fractions in this case are positive. Also, almostany orbit has finitely many reflections within any finite time interval, sincethe boundary components are C ∞ (they are line segments or arcs of a circleof radius r ). Therefore,(3.2) ∞ (cid:88) k =0 (cid:18) ( τ i k +1 + · · · + τ i k +1 ) + 2 r cos ϕ i k +1 (cid:19) = ∞ , where i = 0.Let ˆP denote the space of all non-convex simply connected rational poly-gons. Then, ˆP ⊂ P ref . Theorem 3.3.
For any P ∈ ˆP , there exists r P > such that the physicalbilliard in P is hyperbolic for all r < r P .Proof. Let P ∈ ˆP have n vertices. Assume { v , v , . . . , v n − } and { e = v v , e = v v , . . . , e n = v n − v } are sets of its vertices and edges, respec-tively. Let r k = min {| v k − x | : x ∈ e i f or i (cid:54) = k and i (cid:54) = k + 1 } , where | . | is the euclidean distance in R and k = 0 , , . . . , n − k = 0then i = 2 , , . . . , n − r k is well-defined since it is H. ATTARCHI AND L. A. BUNIMOVICH the minimum value of a continuous function on a compact set. Moreover, r k >
0. If we let(3.3) r P = min { r , r , . . . , r n − } , then the hard ball of radius r < r P will be able to hit all edges of P . There-fore, when r < r P the boundary of the equivalent mathematical billiard hassome dispersing components.In fact, if a radius of the particle is sufficiently large then some partsof the boundary of a billiard table become “non-visible” to the particle.Therefore it does not matter for dynamics what is the exact structure ofthis “non-visible” boundary. Such situation may occur e.g. for polygons [2].As long as a trajectory does not hit dispersing parts, it can be consideredas a trajectory in the rational polygon P (cid:48) that shapes by replacing thedispersing components of the boundary by flat segments (Fig. 4). Moreprecisely, angles θ and α satisfy the equation α = π + θ and θ is commensuratewith π , therefore, α is commensurate with π . A . B θ α . α Figure 4.
Replacing a dispersing part with a line segment.It is well-known that almost all orbits of billiards in rational polygons arespatially dense inside the billiard table [4, 21, 27]. Thus, all non-exceptionaltrajectories in P (cid:48) will hit all edges of P (cid:48) , including those replaced the dis-persing parts of the boundary. This implies that almost all trajectories (fullmeasure in phase space) in the mathematical billiard equivalent to the phys-ical billiard in P will hit at least one dispersing component. Note that afterthe first reflection off a dispersing component, the forward trajectory is notthe same as the one in P (cid:48) . Thus, we cannot use density of almost all orbitsin P (cid:48) to show that the trajectory will hit dispersing parts of the boundaryinfinitely many times.So, there is a full measure subset of points in the phase space such thattheir trajectories hit at least one dispersing component. Let ( x, θ ) be a pointin that subset. By the continuity of the system on initial conditions, there isa neighborhood of positive measure of the point ( x, θ ) such that trajectoriesof all points in that neighborhood hit the same dispersing part as the trajec-tory of ( x, θ ) hits for the first time. Then the Poincare recurrence theorem RIDGE TO HYPERBOLIC POLYGONAL BILLIARDS 7 implies that almost all trajectories in this neighborhood will return and hitthat dispersing part infinitely many times. The convergence of continuedfractions of such trajectories that hit dispersing components follows fromthe Seidel-Stern theorem and (3.2). Hence, for a full measure subset of thephase space of the physical billiard in non-convex simply connected ratio-nal polygons, we have hyperbolicity. Moreover, it implies there are at mostcountable number of ergodic components such that the Kolmogorov-Sinaientropy is positive on each of them [7, 9, 23]. (cid:3)
It follows from Lemma 3.1 and 3.2 that P ref is an open dense set in P .Therefore, being a polygon with at least one reflex angle in P is topologicallygeneric. Theorem 3.4.
There is an open dense subset of P such that the physicalbilliard in the polygons of this subset (when the radius of the hard ball issmall enough) is hyperbolic on a subset of positive measure of their phasespaces.Proof. Let P ∈ P ref and r P > P . Then theboundary of the mathematical billiard equivalent to the physical billiardin P has some dispersing components which are arcs of a circle of radius r < r P . Let, A = { ( x, ϕ ) : F or all x in a dispersing component and ϕ ∈ ( − π , π } . It follows from the definition of A that it is a subset of positive measurein the phase space (one can exclude the exceptional directions which forma measure zero set in the phase space). The Poincare recurrence theoremimplies that almost all points of A will return to A infinitely many timesunder the action of the billiard map. That means almost all trajectoriesof points in A will hit a dispersing part infinitely many times. Then theconvergence of continued fractions of trajectories of almost all points of A follows from the Seidel-Stern theorem and (3.2). Hence a physical billiardin a polygon with a reflex angle is hyperbolic at least on a subset of positivemeasure of its phase space. (cid:3) We conjecture that in fact generically a physical billiard in polygon isergodic for any radius of a moving particle (which is of course not that largethat the particle cannot move within a polygon). To prove our conjectureone instead needs to show that almost all orbits in a physical billiard ina polygon will eventually hit any segment which is a part of any side of apolygon. Hence a “large” physical particle must hit all (rather than one)vertices of a polygon. For instance, if each vertex of a convex polygon getsreplaced by a focusing arc it is possible to prove ergodicity [11]. In this casethe mechanism of defocusing [8] ensures hyperbolicity of such semi-focusingbilliards on entire phase space. The current theory of billiards in polygonsestablishes only that any billiard orbit in a polygon is either periodic or its
H. ATTARCHI AND L. A. BUNIMOVICH closer contains just one vertex of this polygon, which is by far not enough.We are confident though that our conjecture holds.
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E-mail address : [email protected] (L. A. Bunimovich) School of Mathematics, Georgia Institute of Technology,Atlanta, US
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