The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria How High{Dimensional Stadia Look Like Leonid A. Bunimovich Jan Rehacek Vienna, Preprint ESI 455 (1997) June 10, 1997 Supported by Federal Ministry of Science and Research, Austria Available via

2 How High-Dimensional Stadia Look Like Leonid A. Bunimovich School of Mathematics Georgia Institute of Technology Atlanta, GA Jan Rehacek Center for Nonlinear Systems, MS B-258 Los Alamos National Laboratory Los Alamos, NM Abstract We give the armative answer to the long-standing question whether or not the mechanism of defocusing can produce a chaotic behavior in high-dimensional Hamiltonian systems. To do this we prove that billiards in a class of regions in R n ; n > 2, with focusing and at boundary components have nonvanishing Lyapunov exponents. 0. Introduction Systems of a billiard type play a special role in the ergodic theory of dynamical systems. On the one hand, billiards correspond to the rst models that inspired Boltzmann's and Gibbs' creation of the ergodic theory itself. Indeed, the celebrated Boltzmann ergodic hypotheses deals with the gas of hard spheres, i.e. with a billiard. On the other hand, billiards provide the most visible models of nonuniformly hyperbolic dynamical systems. The rst rigorously studied classical dynamical systems with strongly chaotic behavior (that was until recently referred to as a stochastic behavior) were the Hamiltonian systems generated by the geodesic ows on surfaces of a constant negative curvature. Hadamard, Hedlund and Hopf [H, He, Ho1] were responsible for this breakthrough. Anosov [A] and Smale [S] essentially generalized these ideas and introduced much more general classes of hyperbolic dynamical systems, Anosov and Axiom A systems respectively, whose dynamics essentially resembles the one in geodesic ows on surfaces of negative curvature. However, all of these systems are smooth while the classical models (gas of hard spheres) were nonsmooth. Krylov pointed out [K] that a motion of molecules (hard spheres) in the gas of hard spheres resembles very much a dynamics of geodesic ows on surfaces of negative curvature. However, his claim, while being very deep and made well ahead of his time, was extremely vague, and formally it is not proved yet whether or not it was correct. Certainly, everybody believes that it was correct but the Boltzmann hypotheses is not proven yet, even though, the recent breakthrough [KSS1, KSS2] allowed to get its proof for three particles. 1

3 It was Sinai [S1] who started and developed the theory of hyperbolic billiards that is in the heart of the modern theory of nonuniformly hyperbolic dynamical systems. He introduced the class of billiards with a smooth convex inwards boundary. (Such billiards are now called Sinai billiards). Sinai billiards form the class of the "best" nonuniformly hyperbolic systems. However, already in this situation one encounters a series of very essential diculties both technical and the principal ones. (To give an example of such diculties one can mention that the brilliant Hopf's idea of the proof of ergodicity for the smooth hyperbolic systems doesn't work for systems with singularities. Sinai's "main" or "fundamental" lemma demonstrated how one can get around it). One of the applications of Sinai's theory was the proof of the Boltzmann's hypotheses for the gas that contains only two particles (two hard disks). While it is the very special case, it is dicult to overestimate the inuence of this result on the physical community. The striking and "unbelievable" fact that the ergodic hypotheses holds for the system of only two particles, while the Boltzmann's idea always related ergodicity of a system to an extremely large number of particles (degrees of freedom) forced to rethink some basic "physical" philosophy. Sinai's result has changed the "view of the universe" already for several generation of physicists and was the beginning of the triumphal penetration to physics of the ideas of the modern theory of dynamical systems. On the other hand, the classical integrable geodesic ows on the surfaces of positive curvature together with the clasical examples of integrable focusing billiards (circles and ellipses) demonstrated that a focusing (a positive curvature), on contrary to a dispersing, must always help to stabilize dynamics. It was the rm part of the intuition for all mathematicians as well as for all the physicists. Therefore the discovery of the chaotic focusing billiards made in [B2] had the strong impact on the both communities. It demonstrated that there is another mechanism besides dispersing that produces a chaotic motion, i.e. the mechanism of defocusing. It is worthwhile to mention that this discovery, as usually, was made not for purpose, but occasionally. The general philosophy in mathematics, as well as in physics, is that generically strong (robust) phenomena cannot be destroyed by small perturbations. For instance, this kind of idea has been expressed by Hopf [Ho2] who conjectured that the presense of relatively small pieces of positive curvature may not destroy stochasticity of geodesic ows on surfaces of negative curvature. Sinai suggested to the rst author to look at the perturbations of dispersing billiards by small focusing components inserted into a boundary. The corresponding result was proven in [Bu1]. Indeed, under some simple geometric conditions stochasticity of dispersing billiards will be preserved. However, these conditions imply that dispersing boundary can not only be changed onto focusing one at some relatively small pieces but completely removed. The rst class of such regions (containing a surprisingly popular "stadium") was introduced in [Bu2]. Then more general two dimensional focusing billiards with chaotic behavior were studied in [B2-4],[D3],[M1,2],[W2]. The same idea has been applied to the construction of ergodic geodesic ows on a two-dimensional sphere and on a two-dimensional torus [D1-2],[B- G], [O]. However, all these examples studied by all these authors were two-dimensional ones. The principal question, that was around since [B2], remained open: Does the mechamism of defocusing also generate chaotic billiards in higher dimensions? In the recent paper [B-R] the armative answer to this question was obtained in dimension three. The method of proof in this paper essentially used 3D geometry. So, the general question has been just shifted by one more dimension. In the given paper we develop the 2

4 general approach that allows one to prove the existence of nowhere dispersing billiards with nonvanishing Lyapunov exponents in any dimension. It is important to mention that we study the class of billiards that is the natural high-dimensional analog of of the class of billiards introduced in the rst paper [B2] on the new mechanism of chaos in Hamiltonian systems. In particular, our class contains high-dimensional stadia. The structure of the paper is as follows. In the rst section, we introduce the regions in question and review the necessary background from the theory of billiards. In sect. 2 we state some technical lemmas which are necessary to control the curvature evolution during one series of reections in a spherical cap. The third section contains the proof of the main result. 1. Description of the model and the main result We will study billiards in some class of n-dimensional regions (n 2) described below (an example of a typical region is in Fig. 1). The boundary of the region consists of at walls and spherical caps attached to them. We will consider only focusing caps whose angle! 00 < 90. By the angle of the spherical cap we mean the maximum angle subtended by the spherical cap at the center of the sphere. Our aim is to show that by restricting the angular size of the spherical caps, one can achieve focusing that is strong enough to obtain overall divergence of nearby trajectories. C Q ω,, S n φ T -1 x Tx x=(q,v) Figure 1: Billiard trajectory Consider an evolution of an (n? 1)-dimensional innitesimal control surface (also called 3

5 a wavefront) of class C 2 perpendicular to the orbit. The rate at which the neighboring trajectories diverge is measured by the curvature operator (the operator of the second fundamental form) of the surface. To demonstrate that the defocusing mechanism works, we show that if the surface approaches a spherical cap with the positively dened curvature operator, then the whole billiard region can be congured in such a way, that after passing through the spherical cap, the surface will focus "relatively soon" (and thus the curvature operator becomes positively dened again). This property ensures that the mechanism of defocusing discovered in [B2] for 2-D billiard works also for some n-d regions. Since very often we will be working with the n?1-dimensional hypersurface, in the rest of the paper we will adopt the convention m = n? 1. We will mainly use the notation from [B-R]. Let Q R n be a region described above and equipped with a eld of inward normal vectors n(q). Further, let M' be a restriction to Q of the unit tangent bundle of R n. Points in M' have the form x = (q; v), where q 2 Q is the support of x and v 2 T q (R n ). By an n-d billiard we mean a dynamical system in M', generated by the motion of x 2 M 0 along a straight line determined by v with unit speed. When this line reaches the boundary of Q it is reected according to the rule "the angle of reection is equal to the angle of incidence". The angle is measured with respect to the normal n(q). This motion generates a ow on the phase space which we shall denote by S t. As usual, this ow generates the discrete dynamical system with the induced mapping T obtained by the restriction of S t to the boundary (see Fig. 1) M S n. The n-d billiard mapping T preserves the measure d(q; v) = const:(v; n(q)):dq:d!; where dq is the (n-1)-dimensional Lebesgue measure on the boundary of Q generated by a volume and d! is the (n-1)-dimensional (i.e. m-dimensional) Lebesgue measure on the unit sphere. The const is the usual normalizing constant so that (M) = 1. Strictly speaking, the billiard mapping is dened only for a subset (of full measure) of M. Since the detailed information about the singularity set is not necessary for our purposes, we will still call this subset M. The dynamics in the vicinity of a billiard orbit is described by the second fundamental form of the control surface orthogonal to the ow (we will call its matrix the curvature operator) and usually denote its value by (Gx; x). It is necessary to understand how this form changes upon the reection and how it evolves during the free path. Let n(q) be a unit normal vector at the point of reection and let v + and v? be the unit vectors along the directions of the outcoming and incoming billiard orbit respectively. Then v + = v?? 2(n:v? )n and the vectors v + and v? span a plane P in which the whole piece of an orbit lies as long as it doesn't leave the sphere S. This plane is unique and contains any two points of reection in a series of reections from the spherical cap along with the center of the corresponding sphere. The plane P naturally splits the tangent space U of the control surface into two subspaces: U = U p U t ; where U p = U \ P and U t is the (m? 1)-dimensional orthogonal complement to U p in U. The space U has thus two distinguished directions which will be referred to as the planar direction (since it is 1-dimensional) and the transversal (or orthogonal) subspace. 4

6 To obtain formula for the reection, we have to introduce some auxiliary operators (see [S2]), which make up for the fact that the curvature operators before and after reection act on two dierent planes T? and T + as well as the curvature operator of the boundary at the point of reection. Let V be the isometric operator which maps T? onto T + in a direction parallel to the vector n(q) normal to the boundary at q. This operator hence realizes the necessary rotation of the tangent plane so that this is still perpendicular to the direction of reected velocity. Similarly, let W be the operator which maps T + onto T 0 in a direction parallel to the vector v + and let W be the adjoint operator which maps the plane T 0 onto T +. The operator W thus transforms the curvature operator of the boundary onto the tangent plane to the orbit, where it can be added to the curvature operator of the control surface. The explicit formula for this addition is G + = V?1 G? V? 2(v + ; n)w KW where K is a second fundamental form of the boundary at the point of reection q. U p d θ d t U t Figure 2: Decomposition of the tangent space To get an idea, what happens to a control hypersurface upon reection, let us consider a situation on Fig. 2. Here we x one particular unit direction d in the tangent space U. The value of the curvature of in this direction is (G? d; d). To obtain the same directional curvature of after the reection, that is (G + d; d), we write d = d p cos() + d t sin(); where is the angle between d and U p and d p ; d t are normalised projections of d onto U p and 5

7 U t respectively. Then the change in the curvature is 2 + =?? cos 2 ()( rcos() )? sin2 ()( 2cos() ); r where r is the radius of the reecting sphere and is the angle of reection. As a special case of this formula we see that curvature in the planar direction ( = 0) changes just like in the planar case + =?? 2k (1:1) cos() while in the orthogonal subspace it obeys + =?? 2kcos() (1:2): As for the evolution of the curvature operator during the free path, we notice that the principal curvature directions are preserved and the principal curvatures (eigenvalues of the curvature operator) evolve according to the standard "reciprocal" rule (t) = (0) 1 + (0)t where t is the time elapsed from, say x to y. In terms of the quadratic forms we obtain (1:3); G t (y) = G 0 (x)(i + tg 0 (x))?1 ; (1:4) where G t (y) is the second fundamental form acting on the hyperplane U(y) and G 0 (x) is the second fundamental form operator acting on the hyperplane U(x) (perpendicular to the direction of the motion). Now we formulate our main theorem. Theorem 1: Let the contain only spherical caps and at components. Suppose that the angles of all spherical caps are smaller than 90 and that almost every trajectory enters some spherical cap. Then the billiard in Q has non-vanishing Lyapunov exponents, provided that each spherical cap is enclosed in a subregion of Q which has walls perpendicular to the one containing the cap. The size of this subregion should be such that the center of the sphere, whose cap is attached to it, is inside this subregion. 2. Technical results In this section we will prove a series of propositions which we will need in the last section to prove the theorem about non-vanishing of Lyapunov exponents. Recall, that an innitesimal hypersurface perpendicular to the orbit is determined by the curvature operator (the second fundamental form), acting on the tangent space V to this hypersurface. This operator can be regarded as a symmetric m m matrix. In order to study the dynamics in the vicinity of the given orbit we rst state several denitions. 6

8 Denition 1: The eective angle of a billiard orbit in a spherical cap is the angle of the circular arc that the plane of the orbit determines. See Fig. 3 for an illustration. The maximal angle of the spherical cap will be denoted by! 00, while the eective angles of the particular orbit will be denoted by! 0. The letter! will be reserved for an angle between the middle of the rst and last chord of a given orbit. T p Q q φ P n(p) A B ω B, O σ A, Figure 3: Reections in a sphere Denition 2: Two quadratic forms S and S' are said to have substantial tangency, if there exists an (m? 1)-dimensional subspace U 0 U such that for all v 2 U 0 : (Sv; v) = (S 0 v; v): Two innitesimal surfaces perpendicular to the billiard orbit are said to have substantial tangency if their corresponding curvature operators (second fundamental forms) have the above property. Denition 3: A control hypersurface will be called aligned, if one of its principal curvature directions coincides with the planar direction U p U in the tangent space to. Denition 4: A "zone of focusing" of a given spherical cap is a part of the billiard region, which is bounded by the spherical cap, by the at component to which the cap is attached, by at components perpendicular to the one with the spherical cap and nally by a transparent virtual at component, which is parallel to the one with the spherical cap and at the distance R from it. This "bottom wall" will usualy be designated by W and the number R = R(! 00 ; ) will be called the size of a zone of focusing. In this paper, the size of the zone of focusing will be such that the center C of the sphere lies in the bottom wall W. Hence R(! 00 ; ) = cos(! 00 =2). (In Fig. 4, the "top" wall is denoted by T and the "bottom" wall by W ). 7

9 Denition 5: The principal curvatures of the surface at the point of entrance to the zone of focusing will be called entrance curvatures, while the principal curvatures at the point of exit from the zone will be called exit curvatures. Both entrance and exit are through the transparent "bottom wall" W. C T R(ω,,, ρ) ω,, ρ W Q Figure 4: Zone of focusing Remark 1: There are two diculties in the proof of the Theorem 1. One is caused by the fact that in the transversal direction the focusing is much weaker than in the planar direction and orbits must be given sucient time to defocus. The second one stems from the fact, that computation of curvatures of a control surface is feasible only in the case, when one of the principal curvature directions coincides with the planar direction or when the control surface is a piece of a sphere. In this paper, we show that the dynamics of an arbitrary control surface is determined (at least as far as the focusing properties are concerned) by the aligned surfaces. In this sense the behavior of the m-dimensional control surface can be studied by investigating the behavior of m 1-dimensional control curves separately. The same conclusion can also be inferred from the general formalism developed in [L-W]. However, our approach is less abstract and provides a tool to study the quantitative behavior of trajectories in the vicinity of the given orbit. We prove 4 propositions in this section that deal with the evolution of the curvature operator of the control surface. In the rst proposition, we deal with the dynamics in the planar 8

10 direction and in the orthogonal subspace separately. We show, that an innitesimal beam of trajectories which enters the "focusing zone" as diverging leaves the zone again as diverging. For this we will need the perpendicular walls, since the plane of the orbit may cut the spherical cap in a very small angle, which may lead to a weak focusing. The perpendicular walls then ensure that an orbit is then given suciently long time to defocus. In the Proposition 2, we show that a general surface can be approximated by two aligned surfaces which have substantial tangency with, i.e. they share the directional curvature values along an (m? 1)-dimensional subspace. In Proposition 3, we prove that the substantial tangency is preserved during the whole series of consecutive reections. Finally, in Proposition 4, we use the substantial tangency to control the exit curvatures of the general surface by the exit curvatures of the surfaces constructed in Proposition 2. Proposition 1: Consider a billiard orbit having N consecutive reections in a spherical cap, whose (maximal) angle! 00 < 90. Let a zone of focusing around this spherical cap be given, with the "bottom" wall W at the distance of the radius of the sphere. Then every incoming control surface, whose curvature operator is aligned and positively semidenite at the moment of crossing the "transparent" wall W, leaves the zone of focusing as diverging in all the directions, i.e. its curvature operator at the moment the surface leaves through the wall W is positively dened. Proof: First, it is clear from the similarity arguments that it is enough to consider only a spherical cap with the radius 1. In the case of a radius the size of the zone of focusing would be adjusted accordingly and the whole discussion would proceed without changes. Second, since the "spherical" part of the orbit lies in the same plane, we can restrict ourselves to this plane (in Fig. 3 it is the plane ACB, which also contains the points P, Q, p and q). If we follow the incoming orbit beyond A and the outcoming orbit beyond the point B, after some time and possibly after some reections from the surrounding at walls both the incoming and the outcoming orbit will intersect the "bottom" wall W at points A 00 and B 00 respectively. Consider now the plane AOB only. The cap is represented by a circular arc of an angle! 0 (the eective angle), while the "bottom" W of the focusing zone corresponds to a line through the center O of the sphere and parallel to the line representing the top of the zone of focusing. We again extend the in- and outcoming orbits beyond A and B (this time not considering any reections from the surrounding at walls). These orbits intersect the "bottom" line at points A 0 and B 0 respectively. We claim that the total length of the free path from A to A 00 is the same as from A to A 0 and likewise for B. Denote the wall to which the cap is attached by T and its normal vector by n(t ). Since the surrounding walls are perpendicular to the wall T, reections from them do not change the component of the (in-) outcoming velocity in the direction n(t ). Since the time it takes the billiard particle to traverse to the bottom wall W depends only on this component, we could just as well let the billiard particle go through the surrounding walls without any reections at all. But that exactly corresponds to the free path from A to A 0. This is the only part of the proof, where we need perpendicularity to T of the adjacent to its boundary components. In the rest of the proof we will thus consider only a situation in the plane AOB. We begin by stating a lemma, which describes what happens to a curvature of the control surface in the planar and transversal directions. 9

11 Lemma 1: Suppose that the planar and transversal directions are invariant during a series of N reections in the spherical cap of radius = 1 and maximal angle! 00 < 90. Let be the curvature of the control curve in the middle of the rst chord (point A) and 0 the curvature in the middle of the last chord (point B). Then the following formulas hold: (in the planar direction) (in the transversal direction). 0 = 0 = 2N + ; (2:1) cos? sin! sin + cos! cos! + sin!sin ; (2:2) Proof: We prove only the second part of the lemma. The rst one is the well known fact about the planar billiards and can be veried using formulas (1.1) and (1.3). For the proof of the second part we use induction in the number of reections. Recall that we set = 1 and dene! N = N(180? 2); (2:3)! = 180? 2: (2:4) The curvature evolution in the orthogonal subspace is governed by (1.2) and (1.3). First, we will derive a formula for the curvature change from the center of one chord to the next one. Since any chord during one series of reections has the length 2cos, we will need to combine formula (1.3) with t = cos with (1.2) and then with (1.3) again. If we denote the curvature at the beginning of this process by then for the resulting curvature we get 0 =? 2cos? 2cos cos? 2cos 2? 2cos 3 ; which can be simplied to?2cos? cos2 0 =?cos2 + cossin 2 ; and further to?sin2 0 sin =? cos2?cos2 + sin2sin : >From this formula it is clear, that it is advantageous to do the computation with the rescaled curvatures ~ = sin: (2:5) With this convention and using (2.4) we nally obtain ~ 0 = which establishes (2.2) for 1 reection. We now assume the validity of (2.2) for N reections.?sin! + ~cos! cos! + ~sin! ; (2:6) ~ 0 =?sin! N + ~cos! N cos! N + ~sin! N : 10

12 Plugging this expression into (2.6) yields a similar expression with the angle! N +! =! N +1. The formula (2.2) for N + 1 reections is now obtained by returning to the non-rescaled curvatures using (2.5). Remark 2: The formula (2.2) shows that in the transversal direction, the evolution of innitesimal orbit variations (the Jacobi elds) is essentially rotation by the angle!. The formulas at the end of the proof above then correspond to the composition of such rotations. End of the proof of Proposition 1: Working in the plane AOB, we denote the angle AOB again by!, and the angle A 0 OA by. The angle ranges from?90 to 90, while! + > 90 ; (2:7) so that the outcoming ray approaches the "bottom" wall W. The dynamics in the planar direction has been thoroughly studied and from (2.1) the statement of the Proposition easily follows. The only possible complication would occur in case of nearly normal hit of the edge of the spherical cap. In this case the point A 0 would lie "beneath" the wall W and so the curvature at A might be negative and, according to (2.1), so might be the curvature at B. Denote by an angle BB 0 O. Then the angle AA 0 O is 0 < + 2 < 90. If the beam of trajectories that focused at A 0 is to focus again before B 0, then?tan( + 2) + 2 sin cos?tan sin ; where the left hand side is a curvature at B of the family of trajectories focusing at A and the right hand side is a curvature of a family that would focus at B 0. This inequality may be rewritten as tan + 2tan tan( + 2); and is deduced from the convexity of the function tan on (0; 90 ). Thus even in this case the free path between B and B 0 will make the curvature at B 0 non-negative. The dynamics in the transversal subspace U t is uniform in the sense that all the principal curvatures with eigenvectors in this subspace are subject to the same curvature drop (1.2) after the reection. Hence, we can consider the evolution of these curvatures simultaneously and use the formula (2.2) for all of them. Our task is to show that if the curvature (in the transversal direction) at A 0 is non-negative, so is the curvature at B 0. Since (2.2) is a linear fractional transformation (LFT) in with positive derivative and since the free path (from A 0 to A and from B to B 0 ) is also LFT with negative derivative, we infer that the relation between curvature at A 0 and B 0 is also LFT with positive derivative. Thus it is enough to show that image of zero-curvature (at A 0 ) is a positive curvature (at B 0 ) and image of innity is innity. Then we'll have an increasing LFT with positive value at 0 and 1 at 1 and from this we'll conclude that the curvature at B 0 is non-negative. So let us assume that at A 0 we have = 0 and the same curvature is then at A. Thus at B (according to (2.2)) we obtain =? tan!. The length of the free path between sin A0 and A is t = tansin; (2:8) and between B and B 0 t =?tan(! + )sin: (2:9): 11

13 Thus for a curvature at B 0 we get =?1 cotan! + tan( +!) ; which, in view of (2.7), is positive. Now assume, that at A 0 the innitesimal beam of trajectories focuses (i.e. = 1). Then the curvature at A is = 1=(tansin) and at B we get from (2.2) = 1?sin! + cos! tan sin cos! + sin! tan = 1 1? tantan! sin tan + tan! = 1 sin 1 tan(! + ) : This curvature is negative (from (2.7)) and its reciprocal value is exactly the length of the free path between B and B 0, which shows that the innitesimal beam focuses again at B 0. This concludes the proof of the Proposition 1. When the incoming control surface is not aligned, we cannot break the dynamic behavior into planar and orthogonal subspaces separately since this time the principal curvature directions change upon reection. Our rst task will be to construct two auxiliary hypersurfaces that are aligned and both have substantial tangency with. Their alignment allows us to compute their eigenvalues (curvatures), while their substantial tangency with guarantees, that from these computed curvatures one can obtain estimates of curvatures of using the interlacing property of eigenvalues, which we recall below as Lemma 2. Moreover, one of them will majorize and one will minorize it in the following sense. Denote by a and c the majorizing and minorizing hypersurfaces respectively. Since all three hypersurfaces are orthogonal to the orbit at any point, they are all characterised by their curvature operators. They will be denoted by F; G; H for a, and c respectively. Then for all the vectors x 2 U (U is the common tangent space to the surfaces) we have (F x; x) (Gx; x) (Hx; x): (2:10) Lemma 2: Let A and B be two symmetric matrices and let their eigenvalues be denoted by a 1 a 2 ::: a n and b 1 b 2 ::: b n respectively. If the matrix B-A has 1-dimensional range, then their eigenvalues are interlaced, i.e. they satisfy a 1 b 1 a 2 b 2 ::: a n b n in the case A B and reverse inequalities if B A. Proof: The minimax principle claims that the i-th eigenvalue of A is min S max x2s (Ax; x) (x; x) ; where the minimum is taken over all i-dimensional subspaces S of R m. Without loss of generality, we can assume that A B in which case we immediately obtain a i b i for all i = 1; :::; n. 12

14 Next, we observe that every j-dimensional subspace S contains a (j-1)-dimensional subspace S 0 which is orthogonal to the (1-dimensional) range of B? A. Since (Ax; x) = (Bx; x) for x 2 S 0, Hence b j?1 a j. max x2s 0 (Bx; x) (x; x) max x2s (Ax; x) (x; x) : In addition to alignment and substantial tangency with, we want our auxilliary surfaces F and H to have curvatures close to the ones of G. The reason for this is obvious. The more dierent the curvatures are at the beginning, the more they dier later on. In Proposition 2 we show, that the operators F and H can be chosen in such a way, that their curvatures lie "within" the range of curvatures of G. To make this "within" more specic, let us denote by b 1 the smallest curvature of (i.e. the smallest eigenvalue of G) and by b m the biggest one. Proposition 2: Let a semipositively dened curvature operator G be given. Then there exists two aligned positively semidened operators F and H, such that (2.10) holds and both have substantial tangency with G. Moreover, F and H can be chosen so that all their eigenvalues lie in the interval I = (b 1 ; b m ). Proof: We start with the existence of the "minorizing" operator F. Thus, given a real symmetric matrix G, our task is to construct a matrix F, such that i. G F and G? F has rank 1 ii. x p is an eigenvector of F iii. eigenvalues of F lie in I where x p is a unit vector in U p. Recall that the m-dimensional space tangent to all of the three surfaces f, and h is naturally split into U = U p U t. We choose coordinate vectors as follows: for e 1 we take x p and for the remaining m? 1 vectors we take any orthonormal basis in the subspace U t. In these coordinates the quadratic forms F; G and H have matrices f ij, g ij and h ij. We further assume that e 1 is not in the eigenspace of G for if it were then G itself would be aligned and its dynamics (i.e. the dynamics of ) could be studied directly. This implies that g 11 > 0. Indeed, if 0 = g 11 = (Ge 1 ; e 1 ) then e 1 would be an eigenvector, since G 0. Matrices with rank 1 are determined by a vector, say u = (u 1 ; u 2 ; :::; u m ). With this notation the matrix representing the linear mapping v! (v:u)u has a matrix U = (u ij ) = u(i)u(j). Our aim is rst to nd u so that F := G? U satises (i) and (ii). This is easily done by choosing u = (d; g 21 =d; g 31 =d; :::; g m1 =d); (2:11) where d is an arbitrary constant. Since U = G? F is of rank 1 and positive denite (i) holds. Direct computation yields that the rst column of F is (g 11? d 2 ; 0; :::; 0)?. Hence e 1 = (1; 0; :::; 0) is an eigenvector of F, which proves (ii). To show (iii) we have to choose some particular d > 0. Since b 1 is the smallest eigenvalue of G, and e 1 is not an eigenvector of G we obtain g 11 = (Ge 1 ; e 1 ) > b 1. Thus we can choose d = p (g 11? b 1 ) > 0 so that g 11? d 2 = b 1. With this choice 13

15 of d the eigenvalue of F corresponding to e 1 is b 1. Since G F it is clear that eigenvalues of F are bigger than b 1. We now show that with the choice of d as above there are no eigenvalues of F below b 1. Indeed, suppose that there is an eigenvector e and an eigenvalue f (F:e = be) such that b < b 1. We look at the 2D subspace S = span(e 1 ; e). Since b < b 1 the quadratic form (F x; x) (x; x) b 1 for all x 2 S, with equality only for a vector e 1. On the other hand, since b 1 is the smallest eigenvalue of G (Gx; x) b 1 : (x; x) Substantial tangency, however, implies that in the 2D subspace S there must be at least one vector x for which (Gx; x) = (F x; x). From the two inequalities above it follows that this can happen only for x = e 1. But that would imply that e 1 is an eigenvector of G (with an eigenvalue b 1 ) and that possibility we have excluded. One can similarly show that with the above choice of d at least two eigenvalues of F must have the value b 1. We already know that one eigenvalue of F has this value (the one corresponding to e 1 ). Suppose that all the other eigenvalues of F are strictly bigger than b 1. Then let e 6= e 1 be the unit eigenvector of G belonging to b 1. Obviously (Ge; e) = b 1, while (F e; e) > b 1, because F has only one eigenvector belonging to b 1 (and it is not e). Hence at least two eigenvalues of F must be equal to b 1. If we denote eigenvalues of F by a i, we obtain the following conguration a 1 = b 1 = a 2 b 2 ::: a m a m : Of course, with an arbitrary choice of d, the equalities on the left of the above relation become just inequalities. Construction of the majorizing form H is completely analogous. We look for a vector u so that H := G + U satises (i) and (ii). This is accomplished by (note the change in sign) u = (d;?g 21 =d;?g 31 =d; :::;?g m1 =d); (2:11 0 ) where d is again an arbitrary constant. Since H G the eigenvalues of H cannot lie below b 1. Carefully selecting the value d one can again show that there will be two eigenvalues (one corresponding to e 1 ) of H equal to b m. The reasoning is the same as in the case of the "minorizing form" F. That concludes the proof of Proposition 2. In order to illustrate the above construction, let us consider the case m = 3. The quadratic form corresponding to the matrix G can be thought of as an elipsoid with major semi-axes x i (see Fig. 5). Eventually we want to construct an enclosing ellipsoid with one semi-axis pointing in the direction U p and such that it touches the ellipsoid determined by G along a 2-dimensional subspace. We nd easily a sphere H 0, which \encloses" G, however the tangency is, in general, only 1-dimensional (along x 3 in Fig. 5). It is clear that in order to keep the form H aligned, we can \squeeze" the sphere H 0 only along the subspace U t. On the other hand, we want to preserve the tangency, which we have along x 3 and in order to achieve that, we can \squeeze" the sphere H 0 only along the x? 3 subspace. In 3D this results in only one possible direction of squeezing w. In more dimensions, we have to "squeeze" the sphere H 0 14

16 U p x 3 x 2 x 1 U t w H 0 Figure 5: Quadratic form (m? 2) times along suitable directions in order to achieve substantial tangency. This can be formalized in a purely geometrical proof of Proposition 2 which the reader can easily discover for himself. Proposition 3: Substantial tangency is preserved during the free path. Proof: Since during the free path the principal directions stay the same and principal curvatures evolve according to (2.2), we get formula (2.3) for the curvature operator after time t. Note that operators G = G 0 and (I + tg)?1 commute. For the dierence of operators we obtain G t? F t = (I + tg)?1 (G? F )(I + tf )?1 = (I + tf )?1 (G? F )(I + tg)?1 : Now it suces to realize that two operators have substantial tangency if the dierence of their matrices has 1-dimensional range. Knowing this about G? F, we can infer the same about G t? F t directly from the above equality. The substantial tangency is also preserved upon reection, since for each unit direction x we subtract the same quantity from both (F x; x) and (Gx; x) (which represent directional curvatures of surfaces in question. Therefore the substantial tangency is preserved during the whole series of reections in a given spherical cap. The reason for creating this "substantial tangency" in Proposition 2 and showing that it is preserved in a series of reections is that it 15

17 enables us to say, that at each moment the curvature operators satisfy either F G or G F, which we utilize in Proposition 4. Besides, it gives one good estimates of the actual eigenvalues of G in terms of those of F and H (using the interlacing property of the eigenvalues). Let us also mention that, unless the quadratic form G is proportional to unity (and that case is trivial), its eigenvalues are dierent, which entails c 1? a 1 > 0 and c m? a m > 0. Proposition 4: All of the exit eigenvalues of the general surface are bounded by the eigenvalues of a and c, in particular they are all positive. Proof: During the series of reections the eigenvalues a 1 and c 1 (corresponding to the planar direction) evolve according to the known rules for planar billiards. As for the other eigenvalues of F and H (whose eigenvectors lie in the orthogonal subspace U t ), they evolve homogeneously, in the sense that during the reection the same value? 2cos is subtracted from all of them, r while during the free path they evolve independently on each other according to (2.2). After the surfaces a and c arrive at the exit point B 0, they are all non-negative, according to the Proposition 1. Since both surfaces have substantial tangency with, we deduce that F G H at the point B 0. Indeed, by considering 2-D subspace S spanned by unit vector in the planar direction and by a suitable vector in the transversal direction, we obtain (F x; x) < (Hx; x); for all x 2 S: If G < F or H < G, the surfaces a and c couldn't have a substantial tangency with. >From this it follows that the eigenvalues of the quadratic form G (which are the exit curvatures of the surface at B 0 are included between those of F and H, which are all positive. Remark 3: The notion of substantial tangency allows one to make statements about the dynamical behavior of the m-dimensional control surface that cannot be broken down into m separate 1-dimensional cases. This happens whenever none of the principal curvature directions coincides with the planar direction. In this case, the principal curvature directions change with every reection and computation of curvatures according to (1.1)-(1.3) becomes unfeasible. To summarize the results from this section, we now formulate three conditions that we impose on the billiard region, which we assume consists only of the at components and spherical caps (an example of such region is in Fig. 6). Condition A: the angles of all spherical caps are less than the right angle. Condition B: every spherical cap has its own zone of focusing, whose size equals to (or is bigger than) the radius of the spherical cap. Zones of focusing for dierent caps are separated by a positive distance. Condition C: the set of all the phase points x 2 M whose orbit never enters any spherical cap has measure 0. Remark 4: The purpose of "enclosing" each spherical cap in a zone of focusing is to give the outcoming control surface ample time to defocus. By requiring that the respective zones of focusing are separated, we make sure, that control surface always enters the zone of focusing with positive denite curvature operator (second fundamental form). This in turn causes the 16

18 ω,,,, 1 ρ1 1 R( ω, ρ 1 ),, R( ω2, ρ 2 ) ω,, 2 ρ 2 Figure 6: A billiard region Q control surface to enter the corresponding spherical cap with small curvatures and after leaving it to focus while still being in the same focusing zone. The size of the zone of the focusing is such that the original "stadium" is generalized in the most natural way. If we think of the rectangular box (such as the one in Fig. 1) with two spherical caps attached to it on the opposite faces, then the center of the lower sphere should be below the center of the upper sphere. Another purpose of having the zones of focusing enclosed by at components is to destroy the continuous group of symmetries one would obtain if the "stadium" was rotated along the horizontal axis. The reader may also notice that for some regions (e.g. the one in Fig. 1, which has a rectangular cross-section) Condition C is satised automatically. However, for a general position of at walls it is not yet known (although everybody seems to believe that) whether the set of points x whose orbit never enters spherical caps has zero measure. That's why we have postulated Condition C. 3. Lyapunov exponents We shall show that the n-d billiard systems described above have non-vanishing Lyapunov exponents. To achieve this we use the approach via invariant sectors discussed in [W1, L-W]. The key ingredient is to dene a family of cones in the tangent bundle which is invariant (and 17

19 eventually strictly invariant) with respect to the billiard map. Before we state the theorem, we would like to review a few facts from the geometry of tangent vectors and from the symplectic geometry. It is customary to relate tangent vectors for billiard systems to innitesimal families of trajectories. More precisely, consider a point x = (r; ) 2 M (as in Fig. 7), determining a dashed billiard orbit (both r and have m components). A vector x 0 = (r 0 ; 0 ) 2 T x M can naturally be related to a family of orbits o() = (r(); ()) = (r + r 0 ; + 0 ): Note that this family represents a curve in the phase space, satisfying o(0) = x and o 0 (0) = x 0. Hence this family is a natural representative of a class of equivalent curves from the usual denition of a tangent vector. φ r r+dr dφ x φ x r Figure 7: Tangent vectors However, representing tangent vectors as families of trajectories originating from the billiard boundary has one formal drawback. For the purpose of expressing the behavior of nearby trajectories it is convenient to use the curvature of the wavefront corresponding to the family o(). Note that two tangent vectors which dier only by a scalar multiple give rise to families with the same curvature. The quantity d, which is the natural candidate to look at, is related dr to this curvature through a factor of cosine of the angle of reection. For this reason we will consider the new arc-length parameter, which can be thought of as measuring the distances in the plane perpendicular to the orbit rather than on the billiard boundary. Consider an arbitrary point x = (q; v) 2 M. By simply adding the arc-length parameter in the direction of the motion, we have a complete set of coordinates on T x (Q S m ) for r 0 18

20 any point of the (continuous) billiard orbit. Since the dynamics in the direction of the motion is trivial, this coordinate is usually suppressed. The remaining coordinates parametrize the plane perpendicular to the orbit at any point of Q, which includes the boundary points. This can be thought of as taking the tangent space to Q S 2, quotiened by the direction of the orbit. In the remainder of this section T x M will always mean this perpendicular subspace of the tangent space at each point. Thus T x M can be dened also for non-boundary points x. Now we will take a closer look at the tangent vectors and the curvatures of wavefronts dened by the associated families o(). First let us recall that the second fundamental form of any smooth surface is a quadratic bilinear form G, expressing the change in the normal vectors of neighboring points. In terms of our coordinates, G can be expressed as G = 0 d 1 dr 1 d 1. d m dr 1 dr 2 : : :.... d m dr 2 : : : Suppose that we x a point x = (r 1 ; :::; r m ; 1 ; :::; m ) 2 M and a tangent vector x 0 = (r 0 1 ; :::; r0 m ; 0 1 ; :::; 0 m ) 2 T xm. This vector denes a family of trajectories o(), which can be also thought of as an innitesimal curve, (perpendicular to the orbit). We will describe below how to express the curvature of this innitesimal curve using the second fundamental form. Denote r 0 = (r 0 1; :::; r 0 m) and 0 = ( 0 1; :::; 0 m). Then the second fundamental form of a surface maps the arc-length vector onto the angular vector 0 = Gr 0 : Since the curvature of a surface in the direction of a unit vector u is given by (Gu; u), taking u = r 0 =jr 0 j allows us to compute the curvature of the family o() as d 1 dr m. d m dr m 1 C A : = 0 :r 0 jr 0 j 2 : (3:1) Since the curvature depends only on the direction we can always rescale the vector x 0 so that r 0 is a unit vector. Since the dynamics around the given orbit is best described by a local hypersurface, orthogonal to the orbit, rather than by a curve, we will consider such orthogonal hypersurfaces. They correspond to m-dimensional subspaces in the tangent space, spanned by m independent vectors x 0 i = (ri; 0 0 i). However, not every m-d subspace in the 2m-D tangent space corresponds to an innitesimal perpendicular surface. In order that span(x 0 1; :::; x 0 m) corresponds to an innitesimal surface, it is necessary and sucient that for all i; j = 1; :::; m. r 0 i: 0 j = r 0 j: 0 i (3:2): This is just a condition for the symmetricity of the curvature matrix G. If we think of R 2m as a symplectic space with a standard symplectic form, then the equation (3.2) becomes just (x 0 i; x 0 j) = 0. Hence the innitesimal surfaces, perpendicular to the orbit can be identied with the Langrangian subspaces of R 2m, i.e. with planes that are skew-orthogonal to themselves ((x 0 ; y 0 ) = 0 for any two vectors from that plane). 19

21 Before we prove the main theorem, let us introduce the notion of sectors in the tangent space (for more detailed treatment see [L-W]) and recall some elementary facts about them. Let V 1 ; V 2 T x M be two transversal Lagrangian subspaces, i.e. every vector in w 2 T x M can be uniquely written as w = v 1 + v 2, where v i 2 V i. This decomposition allows one to dene a quadratic form Q(w) =!(v 1 ; v 2 ) on T x M. Recall that (x 0 ; y 0 ) = r 0 : 0? s 0 : 0 ; where r 0 ; s 0 ; 0 ; 0 2 R m and x 0 = (r 0 ; 0 ); y 0 = (s 0 ; 0 ) 2 R 2m = Tx M (" =" denotes an isomorphism between the linear spaces). Given V 1 and V 2 we can dene a sector (cone) C = C(V 1 ; V 2 ) = (w 2 T x M; Q(w) 0): (3:3) The interior of the sector is then dened as the set of vectors on which the quadratic form Q is strictly positive. Since the denition (3.3) is dicult to work with, we will now evaluate the quadratic form Q explicitly for a particular choice of the Lagrangian subspaces V 1 and V 2. Namely, V 1 = f(r 0 ; 0); r 0 2 R m g; (3:4) V 2 = f(0; r 0 ); r 0 2 R m g: (3:5) It is clear that these two subspaces are Lagrangian and that they are transversal, i.e. R 2m = V 1 V 2 ("" stands for a direct sum). These subspaces correspond to innitesimal surfaces, one of which is at and one is focusing (i.e. with innite curvature) and the corresponding sector is called the standard sector (for more detailed treatment, see again [L-W]). With this choice the quadratic form becomes Q(x 0 ) = r 0 : 0 : It is clear that if Q(x 0 ) > 0 (or Q(x 0 ) 0), then we can nd an innitesimal surface with positive denite (semidenite) curvature operator, such that a vector x 0 lies in the Lagrangian subspace that corresponds to it (the reader can nd more details in [B-R]). On the other hand, every innitesimal surface with a positive denite (semidenite) curvature operator lies (strictly) in the standard sector C(V 1 ; V 2 ). Thus invariance of sectors can be fully described by means of curvature operators of the innitesimal surfaces perpendicular to the orbit. We can now nish the proof of the main theorem of this paper (see Sect. 1). Theorem: The billiard map T for the region Q satisfying the conditions (A), (B) and (C) has non-vanishing Lyapunov exponents for almost every x 2 M. Proof: For a point y i = (R i ; u) 2 M (see Fig. 8) for which the trajectory is dened we will construct the sectors (cones) rst for certain points inside the billiard region and then translate them using the dierential of the ow back to the boundary. We dene these sectors roughly as those representing surfaces which have positive curvatures at the entrance of any focusing zone, which the region Q may contain. More precisely, given a point R i, let us denote by A i the point in the middle of the chord immediately before the rst reection in the next series of reections in any spherical cap that belongs to Q. Since each spherical cap is enclosed in a zone of focusing, each point A i has a corresponding point A 00 i at which the orbit going through A i enters the focusing zone (see the 20

22 u R i O A,,, i A i Q Figure 8: Invariant cones construction beginning of the proof of Proposition 1). At these points A 00 i we will dene the sectors, which we then move to other points. The fact, that we can nd such A 00 i 's for a set of points of full measure follows from Condition C and from the Poincare Recurrence Theorem. We denote the unit velocities at the points A 00 i by v i and set x i = (A 00 i ; v i). For the given billiard orbit we denote the map carrying x i to x i+1 by s (thus s(x i ) = T t i (x i ) = x i+1 for a suitable time t i ) and the corresponding dierential that acts from T xi M to T xi+1 M by S. Since x i 's are not points of the boundary, we must explain what we mean by T xi M. It is an orthogonal complement of the velocity vector v i in the (2m + 1)-D tangent space T xi Q S m. Since this space is 2m-D and plays the same role as T x M we keep this notation at points x i. Thus we factor out the direction of motion and look only at the dynamics in the orthogonal complement of this direction. By considering rst the dynamics between the conguration points A 00 i, we establish the non-vanishing of Lyapunov exponents for the rst return map (with respect to the focusing zones). The non-vanishing of the Lyapunov exponents for the case of the billiard map itself then follows from the standard argument (see [W1]), i.e. the Lyapunov exponents of the map T and of the "rst return map" s (between x i 's) are proportional; the constant of proportionality being the average of the return time t i (T t i x i = x i+1 ), whose existence is guaranteed by the Ergodic Theorem. Let x i be a phase point that corresponds to A". We dene the cone C(x i ) at a point x i by a standard sector described by (3.3), (3.4) and (3.5). 21