Slow mixing billiards with flat points

Transcription

1 Slow mixing billiards with flat points Hong-Kun Zhang 1,2 Abstract We study mixing rates of a one-parameter family of semi-dispersing billiards whose curvature vanishes at finitely many points (flat points) on its convex boundaries. There are infinitely many periodic trajectories through zero-curvature points, which trap nearby trajectories and slow down the mixing rate. It is shown the correlation function of the collision map decays polynomially with order O(1/m), which does not depend on with the parameter of the family. Introduction Billiard system is a mechanical system where a point moves freely at unit speed in a domain Q and reflects off its boundary Q (the wall) by the rule the angle of incidence equals the angle of reflection. It preserves a uniform measure on its phase space, and the corresponding collision map generated by the collisions of the particle with Q preserves a natural absolutely continuous measure on the collision space. Many classes of planar chaotic billiards have been proven to be hyperbolic, and some ergodic, mixing, and Bernoulli, see [11, 1]. However, ergodic and mixing systems (even Bernoulli systems) may have quite different statistical properties depending on the rate of mixing. Many Received date: September 25, Department of Mathematics and Physics, North China Electric Power University, Baoding, Hebei, PRC, ; 2 Department of Mathematics, Northwestern University, Evanston, IL,

2 mixing and Bernoulli systems have slow (polynomial) mixing rates, which cause weak statistical properties. Even the central limit theorem may fail. Such systems exemplify a delicate transition from regular behavior to chaos. For this reason they have attracted considerable interest in mathematical physics during the past 20 years, see [4, 12, 13]. The rates of mixing in chaotic billiards is rather difficult to establish, because the dynamics have singularities, which aggravate the analysis. A general approach to the studies of billiards with slow mixing rates was developed by N. Chernov and the author in 2005 [7], it is based on recent Young s results [Y98, Y99] and [Mar04]. We consider a family of semi-dispersing billiards with finite flat points on convex boundary and assume that there exists a trajectory with infinite horizon and touches the convex boundary only at the flat point. In addition, there are a sequence of periodic trajectories converging to the trajectory with infinite horizon. Notice the vicinity of these orbits also acts as a trap where hyperbolicity may remain weak for arbitrarily long times, thus the mixing rate is slow and should be slower than semi-dispersing billiards (since they only have singularities of infinite horizon). This is a class of chaotic billiards hardly ever been investigated before. Our main result, stated precisely in the next section, is that, for the class of billiards we consider here, if β (2, 1 + 2), the correlations for the collision map decay as O(m 1 ). Surprisingly, the decay rate does not depend upon the parameter, β. Furthermore, we investigate in detail the singularities for this class of billiards for all β > 2. And we conjecture that the mixing rate should remain the same even for β > Since our method depends upon a strong condition (the One-step Expansion Estimates) developed in [7], which we believe can be relaxed. 1 Statement of Results For simplicity we assume that there are two symmetric flat points and there are trajectories with infinite horizon tangential to Q only at the flat points. More precisely, let Q be a domain in the stripe R = {(v, w) : 1 v 1} bounded by the convex curves Γ ± given by w = v + a β + 1, and w = v a β 1 where β (2, ) and a (0, 1), see Figure 1. Note that the curvatures 2

3 of Γ ± do vanish at the flat points p = (a, 1), q = ( a, 1), and there exist trajectories tangential to the boundary at flat points only and have infinite horizon. The power β is the parameter of the family of billiard tables. Notice there are infinitely many periodic trajectories between flat points. Γ + w Q v Γ Figure. 1: Semi-dispersing billiards with flat points. The dynamics preserves a uniform measure on the the unit sphere bundle of Q. Let M = Q [ π/2, π/2] be the standard cross-section of the billiard dynamics, we call M the collision space. Denote π : M Q be the projection map. Canonical coordinates on M are r and ϕ, where r is the arc length parameter on Q and ϕ [ π/2, π/2] is the angle of reflection, see Figure. 2. Notice here we define r coordinate such that the flat point p has r coordinate 0. n r < 0 π 2 p ϕ r > 0 π 2 Figure. 2: Orientation of r and ϕ The first return map F : M M is called the collision map or the billiard map, it preserves smooth measure dµ = const. cosϕdr dϕ on M. For any f, g L 2 µ (M), correlations are defined by (1.1) C m (f, g, F, µ) = (f F m ) g dµ f dµ g dµ M M M 3

4 It is well known that F : M M is mixing if and only if (1.2) lim m C m(f, g, F, µ) = 0, f, g L 2 µ (M). The rate of mixing of F is characterized by the speed of convergence in (1.2) for smooth enough functions f and g. We will always assume that f and g are Hölder continuous or piecewise Hölder continuous with singularities that coincide with those of the map F k for some k. For example, the free path between successive reflections is one such function. We say that correlations decay exponentially if for some d 1 > 0 and polynomially if C m (f, g, F, µ) < const e d 1m C m (f, g, F, µ) < const m d 2 for some d 2 > 0. Here the constant factor depends on f and g. Next we state our main results. Theorem 1. For the family of billiards defined above, if β (2, 1 + 2), then the correlations (1.1) for the billiard map F : M M and piecewise Hölder continuous functions f, g on M decay polynomially: (1.3) C m (f, g, F, µ) const m 1, Remark. In the limit β 2, the boundary with flat points curves up and Γ ± approach to strictly dispersing boundaries with nowhere vanishing curvature; then the billiards corresponding to semi-dispersing billiards with infinite horizon and so the correlations decay polynomially with order O(m 1 ) as proved in [7, 9]. On the other hand, as β gets no larger than 1 + 2, the correlations still decay with the same rate. Of course the result is no longer true as we take the limit for β, since then the billiard table has all flat boundaries, and there is no chaos at all. Thus this family of billiards demonstrate a transition from chaos to regularity. Although we do not study the decay of correlations for the case when β 1 + 2, but we investigate the dynamics of the class of billiards for all β > 2 in Section 2. And we conjecture that the mixing rate is the same for all β > 2. From now on, we will denote by C > 0 various constants whose exact values are not important. 4

5 2 Dynamical properties of reduced map Let M + denote the part of the collision space corresponding to convex walls Γ ± Q, i.e. M + = {(r, ϕ): r Γ ± }. One divides M + into countably many sections (called homogeneity strips) defined by and H k = {(r, ϕ) M + : π/2 k 2 < ϕ < π/2 (k + 1) 2 } H k = {(r, ϕ) M + : π/2 + (k + 1) 2 < ϕ < π/2 + k 2 } for all k k 0 and (2.1) H 0 = {(r, ϕ) M + : π/2 + k 2 0 < ϕ < π/2 k 2 0 }, here k 0 1 is a fixed (and usually large) constant. Then, a stable (unstable) manifold W is said to be homogeneous if its image F n (W) lies either in one homogeneity strip of M + or in M \ M + for every n 0 (resp., n 0). It is shown in [3] that a.e. point x M has homogeneous stable and unstable manifolds passing through it. We note that the dynamics for this class of billiards are very similar to those of semi-dispersing billiards with infinite horizon. But because the latter has been popular for a long time, its dynamical properties have been investigated in great detail (see, for example in [3, 5]. These include sharp estimates on distortion bounds, conditional densities on unstable manifolds, and the Jacobian of the holonomy map. For the billiards we consider here, such estimates should still hold, or at least in a weaker form. We plan to publish separately a detailed investigation of the these properties for the class of billiards with flat points. We remove from M + the boundaries H k, thus making M + a countable disjoint union of the open homogeneity strips H k s. Accordingly, the images (preimages) of H k need to be added to the set S 1 (resp. S). We call them new singularities, and refer to the original sets S and S 1 as old singularities. Define M = M + and F : M M the reduced map of F on M. For the reduced map F, the reduced natural smooth invariant measure of µ on M is in fact an SRB measure [12]. It is easy to see that F has one type of singularity that make traps for trajectories the infinite horizon singularity. 5

6 We say a point has infinite horizon (or unbounded horizon ), if the free paths between successive collisions may be arbitrarily long in the unfolding space. Notice there are 4 IH-points based at the two flat points on Q. By symmetric property of the billiard table, the dynamics in the vicinity of each IH points are similar. For simplicity, we only concentrate on the IH point z + = (0, π/2) bases at flat point p, and denote z = (0, π/2). For any x M, denote γ x to be the forward trajectory of x. Also notice that in the neighborhood of z +, there exists a sequence of periodic trajectories {γ ym } starting from flat point p with period 2m, m N. In the unfolding space, trajectories {γ ym } only touch Q at flat points, and they converge to the trajectory γ z+. Thus trajectories of points in the vicinity of stable manifold wm s of y m may experience arbitrary number of collisions near almost flat boundaries, which makes the hyperbolicity very weak there. We take a close look at the neighborhood of z + in M. The discontinuity curves of the map F in the vicinity of z + are of two types. There is a long curve S 0 through z + running into M. In addition, there is a sequence of curves corresponding to discontinuities of return time function defined (2.2) R(x; F, M) = min{r 1 : F r (x) M} for x M. These curves S m, m N, running between S 0 and the border M, converge to z + as m goes to. More precisely, if denote M m to be the domain bounded by the curves S m, S m+1, S 0 and M, which is commonly called an m cell. Then for any x M m, the return time function R(x; F, M) = m+1. The dimensions of the components of this structure are indicated on Figure. 3. Notice that S m contains only points whose F m images tangentially touch the convex boundaries of Q. Thus for any x = (r, ϕ) S m, we have (2.3) π 2 ϕ β r r β m On the other hand, S 0 contains points whose F images tangentially touch the convex boundary of Q, and any x S 0, r > 0. Thus for any x S 0, (2.4) π 2 ϕ β rβ rβ. 6

7 ϕ m 1 1 β z + m 1 β F 1/m M m m 1 β β F(M m) W s S 0 F(W s ) r z Figure 3: Singularity curves of F near an IH-point z + and structure of near z. F M Combining (2.3) and (2.4), for any x = (r, ϕ) S 0 M m, then 1 (2.5) r 1 β m, π 2 ϕ 1 β, m N. m β 1 It is not hard to get the structure of FM m, which is similar to M m, but its long sides have negative slope. In fact from the above calculations, we can furthermore calculate intersection points of boundaries of M m and FM m. By the involution, points (r, ϕ) F(M m ) correspond to (r, ϕ) M m. Thus the domain F(M m ) is symmetric to M m about the horizontal line ϕ = 0. Figure 3 shows the structure of singularity lines of the map F near the IH point z +, in the r, ϕ coordinates. Corollary 2. There exist positive constants c 1, c 2 and C, such that for any x M m, F 1 x M n, with (2.6) c 1 β m β 1 C n c 2 β 1 m β + C Notice for any m N, there exists a point y m M m with period 2m, whose trajectory collides only with points of zero curvature. Furthermore, each periodic point y m has a stable manifold wm s which attracts point whose trajectory will experience a large number of collisions with points of arbitrary small curvature. Thus the hyperbolicity is highly nonuniform even in each m cell M m. 1 Notation A B means C 1 < A/B < C, where C = C(Q) > 1 is a constant. 7

8 Q q3 m q2 m y 1 γ y2 γym Q q4 m q1 m Figure. 4: Trajectories with collisions only on flat points in the unfolding table To guarantee uniform hyperbolicity of the reduced map, we need to get rid of small neighborhood of y m in each M m. Fix a constant ε > 0 and for any m N, define a small set U m = ([ ε m, ε m ] [ π/2, π/2]) M m, where ε m = εm 1 2 β. By the remark above Corollary 2, we know F m (M m ) U m. Define (2.7) M = {x M : x M m \ (U m wm s ), m N}), i.e. we remove from M + narrow windows that contains points based at flat points, see Figure. 4. Denote F to be the reduced map of F on M, and C m = M m \ U m. Notice for any point x U m, we have (2.8) cosϕ 1 m. Clearly the map F preserves a reduced mixing SRB measure µ. Furthermore, any unstable manifold in M has distortion bound, bounded curvature and absolute continuity properties. For any m N, consider a short unstable curve W C m, let x = (r, ϕ) W and v = (dr, dϕ) be a tangent vector at x of W. We put (2.9) B(x) = 1 cosϕ ( dϕ dr + K(r) where K(r) is the curvature of the boundary Q at the point of reflection, r. We define a special p-metric on vectors v = (dr, dϕ) by (2.10) v p = cos ϕ dr. 8 )

9 Put x 1 = (r 1, ϕ 1 ) = Fx and v 1 = (dr 1, dϕ 1 ) = DF(v). It follows from the mirror equation of geometric optics, see [2, 3], the expansion factor is (2.11) Λ(x) p = v 1 p 1 + τ(x)b(x) 1 + τ(x)k(r) v p cosϕ where τ(x) is the collision time for x under F and clearly is bounded from below by 2m. Notice for x C m, (2.12) K(r) β(β 1) r β 2 Cm 1, and by (2.3), we have (2.13) cosϕ β r β m c 1m 1 β β. Combine (2.11) we get Λ(x) p c 2 m 1 1 β. Now consider the expansion factor in Euclidean metric dv 2 = (dr) 2 + (dϕ) 2. Note that Λ(x) p cosϕ τ(x)k(x) c 3 Thus (2.14) Λ(x) = Λ(x) p cos ϕ cosϕ 1 By (2.6), for x C m with Fx C n, thus 1 + ( dϕ 1 dr 1 ) 2 1 c ( dϕ cosϕ dr )2 1 cosϕ 1 c 5 m (1 β β )2, Λ(x) cm (1 1 β )2. Thus F is uniform hyperbolic on M. The cutting of a U m out of M m results in new types of singularity curves of F in each m cell M m. Figure 5 shows the structure of singularity lines of the map F in M m, in the r, ϕ coordinates. There are bold steeply decreasing curves c m and c m terminating on S 0 or M, consist of the points in M m whose trajectories hit the point q m 2 or q m 3 9

10 z + s m,k U m c m S 0 S m+1 S m s m,k w s m Figure. 5: Singularity curves of F in C m. of Q under the map F m. The stable manifold wm s crosses c m, consists of points whose trajectories converge to γ ym. The dashed part of wm s does not enter the window immediately, but will do so in one iteration. The singularity set S C m of the map F consists of two types of infinite sequences of singularity curves {s m,k } and {s m,k }, k N, running between S m and S m+1, which correspond to the discontinuities of the function R(x; F, M) in C m. The region blow wm s but above c m consists of points whose trajectories enter the window and manage to move through it crossing γ ym in the unfolding space Q. This region is divided into a sequence of almost parallel strips by {s m,k }, k N. Let C m,k be the strip bounded by s m,k, s m,k+1, c m, c m, S 0 and U m. The region above wm s but below c m consists of points whose trajectories enter the window but turn back without reaching γ ym in the unfolding table Q. This region is divided into a sequence of almost parallel strips by {s m,k }, k N, see Figure. 5. Denote by C m,k the strip bounded by s m,k, s m,k+1, c m and U m. Both C m,k and C m,k consist of points experiencing exactly km collisions with the boundary of Q before returning to M. In order to determine the rates of the decay of correlations we need certain quantitative estimates on the measure of the regions C m,k and C m,k and on the factor of expansion of unstable manifolds W C m,k and W C m,k under the map F. Proposition 3. Unstable manifolds W C m,k and W C m,k under the map F by a factor Λ m,k m 1 1 3β 2 β k β 2. are expanded The proposition will be proven in the Appendix using a similar approach as in [8]. 10

11 Theorem 4. Both C m,k and C m,k have measure of order O(m 3+ 1 β 1 k 2 3β β 2 ). Proof. Due to the time-reversibility of the billiard dynamics, the singular curves in F M have a similar structure. Furthermore, F maps each region C m,k onto a symmetric region made by the singular curves in FM. Long sides of C m,k are transformed into short sides of F(C m,k ), while short sides of C m,k are transformed into long sides of F(C m,k ). Unstable manifolds W C m,k (which are short increasing curves in the r, ϕ coordinates) are mapped onto long unstable curves stretching across F(C m,k ) completely. Let h m,k denote the maximum height of the region C m,k. Let W be the longest unstable curve in C m,k, i.e. its left end point is very close to U m. Thus the slope of W can be approximated by dϕ dr = rβ 2 C m. Thus h W m Since the length of F(C m,k ) is O(m 1/β ), the factor of expansion of unstable manifolds W C m,k is (2.15) Λ m,k Thus (2.16) h m,k 1 m 1/β W. 1 m 2 k 3β 2 β 2 A similar analysis applies to the region C m,k. According to (2.16), the measures of both C m,k and C m,k are of order O(m 3 k 3β 2 2 β ), notice we use the fact that the density in Cm,k or C m,k is of order O(m 1 β β ) and the length of such a cell is of order O(m 1 β ). The estimations outlined in above proof is the result of rather straightforward geometric calculations, so we omit unnecessary details.. 11

12 3 Exponential decaying property of the reduced map when β (2, 1 + 2) Here we use a simplified method to prove exponential decay of correlations for our reduced billiard map. It is mainly based on recent results in [13, 5, 7]. Firstly, we divide the singularity curves S of the map F: a finite number of primary curves and countably many sequences of secondary curves, each sequence converges to a limit point or a curve in S. More precisely, we declare all the short singularity curves S m with m > m 0 secondary, where m 0 is sufficiently large and will be specified below. All the other old singularity curves of F are declared primary. The new singularities consist of the preimages of the boundaries of the homogeneity strips H k and all s m,k s m,k. In particular, both S m and w m are limit curves for different sequences of new singularity curves. All the new singularity curves are also declared secondary. Notice dist(y m, S m ) dist(y m, S m+1 ). So we can modify one of the homogeneity strips, such that the region of C m between c m and c m is completely contained in one homogeneity strip H km. It is known that the complexity K m of the primary singularities of the map F m grows at most linearly, i.e. (3.1) K m C 1 + C 2 m, where C 1, C 2 > 0 are constants (see [2], Section 8). This means that no more than C 1 + C 2 m singularity curves of S m meet at any one point x M. Lemma 5. (One-step expansion estimate) Assume β (2, 1 + 2), let W be a short unstable manifold that crosses no primary singularity curves but finitely or countably many secondary singularity curves. Then (3.2) lim inf δ 0 0 sup Λ 1 i < 1, W : W <δ 0 where the supremum is taken over unstable manifolds W intersecting no primary singularity curves and Λ i, i 1, denote the minimal local expansion factors of the connected components of W \ S under the map F. Proof. Let v be an unstable tangent vector based at x and v m = D x F m (v) its image. Let τ m be the geodesic distance on Q between x m = F m x and i 12

13 x m+1 = F m+1 x, then by (2.11) and the first part of (2.14), the expansion factor of v, in the Euclidian metric, is given by the formula (3.3) Λ m = v m+1 v m C τ mk(r m ) cos ϕ m+1 C mk(r m) cosϕ m+1. Suppose first that a short unstable manifold W crosses only preimages of boundaries of H k converging to a primary old curve. Notice in each C m, there are no other singularity curves between S m+1 and c m. Fix an integer m 1, for any m m 1, then W C m is divided into countably many pieces W k = W F 1 (H k ). The expansion of W k under F is bounded below by Λ k Ck 2, where C > 0 is a constant. Thus, Λ 1 k k 0 k 0 (Ck 2 ) 1 2 C 1 k 1 0, which can be made < 1/2 by choosing k 0 large enough, say k 0 > 4 C 1. Next, let a short unstable manifold W intersect no singularity curves S m but only infinitely many new singularity curves s m,k (or s m,k ) converging to wm s as k goes to. Notice every unstable manifold W M is a smooth monotonically increasing curve in the r, ϕ coordinates. Hence for every k 1 the intersection W C m,k is at most one curve, and the same is true for W C m,k. If W crosses the separating line ws m, then it intersects C m,k and C m,k for all k k δ, where k δ grows to as W = δ converges to 0. Then 1 2 3β < const β k β 2 < const m β β k 2 β k Λ 1 k k=k δ m 1+ δ, which is less than 1 for all sufficiently small δ > 0. If W does not cross wm s, but crosses s m,k or s m,k, the analysis is similar. At last, let a short unstable manifold W intersect some secondary old curves S m with m 0 m m 2, and near each of them it intersects infinitely many new singularity lines. Notice this can only happen in the upper region left to the vertical line r = 0. For each m m 0 denote by W m the piece of W between S m and S m+1. Then any x = (r, ϕ) W m, with m < m 2, we have r cm 1 1 β. Notice for Wm2, its right end point may lie on the boundary of U m2, so for any x = (r, ϕ) W m2, we have r εm 1 2 β. Thus by the first inequality in (2.12), { (3.4) K(r) c 1 r β 2 c 2 m β 2 1 β, if m < m2 ; c 3 m 1, if m = m 2. 13

14 For k k 0 (k k m ), let W m,k = W m F 1 (H k ) denote the connected components of W \ S. By Corollary 2, It is easy to observe that the image F(W m ) only intersects homogeneity strips H k with k cm β 1 2β, where c > 0 is a constant. By (3.3), the expansion of W m,k under F is bounded below by Λ m,k CmK(r)k 2. Thus we have m 2 ( m=m 0 β 1 k=cm 2β Λ 1 m,k ) m2 ( m=m 0 m 2 β 1 k=cm 2β m=m 0 2 C 1 c 1 m 10 C 1 χ 1 β 2 2β 1 2β(β 1) m0. (Cm 1 β 1 k 2 ) 1 ) β β(β 1) By assumption on β, for β (2, 1 + 2), β 2 2β 1 < 0, thus the last expression in above inequalities can be made < 1/2 by choosing m 0 large enough. In [5, 7], the following lemma was proved. Lemma 6. If the reduced map F satisfies (3.1) and (3.2), then there is a horseshoe 0 M such that (3.5) µ ( x M : R(x; F, 0 ) > m ) Cθ m m N, for some θ < 1, where R(x; F, 0 ) is the return time of x to 0 under the map F. Thus the map F : M M enjoys exponential decay of correlations. Hence we conclude that for β (2, 1 + 2), the return map F : M M has exponential mixing rates by above Lemma. 4 Proof of the main Theorem 1 A general strategy for estimating the correlation function C m (f, g) for systems with weak hyperbolicity was developed in [7, 9]. The tower in M can be easily and naturally extended to M, thus we get a the Young s tower with the same base 0 M; and a.e. point x M again properly returns to 0 under F infinitely many times. Consider the return 14

15 times to M under F for x M. According to Figure (3),the cell C m has width in ϕ dimension O(m 2 ), height in r dimension O(m 1/β ), and the density of the measure µ on C m is O(m (1 β)/β ), hence (4.1) µ(c m ) = O(m 3 ). Also notice 1 m n k n m It is immediate that for any large n, (4.2) µ(x M : R(x; F, M) > n) m n (4.3) Equivalently, [µ(c k,m ) + µ(c k,m)] < n 2 µ(c m ) + const n 2. 1 m n k n m [µ(c k,m ) + µ(c k,m )] (4.4) µ(x M : R(x; F, M) > n) const n 1 n 1 The equivalence of (4.4) and (4.2) is proven in [7]. For every m 1 and x M denote Let r(x; m, M) = #{1 i m : F i (x) M} A m = {x M: R(x; F, 0 ) > m}, B m,b = {x M: r(x; m, M) > b ln m}, where b > 0 is a constant to be chosen shortly. By (3.5), we know that Choosing b = 2/ ln θ, then µ(a m B m,b ) C m θ b lnm. (4.5) const m θ b ln m const m θ 2 ln θ ln m = const m 1. The set A m \ B m,b consists of points x M whose images under m iterations of the map F return to M at most b ln m times but never return to the base 0 of Young s tower. Our goal is to show that µ(a m \ B m,b ) = O(m 1 ). 15

16 Lemma 7. For any x C m FC n with n [c 1 m β 1 β, c2 m β β 1 ], the transition probability from C m to C n is µ(f(c n )/C m ) = { where c is the normalized factor. cmn 2β 1 1 β + O( 1 ), if n m; m cn 2 m β 2 β 1 + O( 1 ), otherwise, Proof. By (2.3) and (2.4), for any x = (r, ϕ) C m F(C n ) with n [c 1 m β 1 β, c2 m β β 1 ], π/2 ϕ = C 1 1 m 1 n, rβ 1 = C 2 ( 1 m + 1 n ) It is enough to consider the case when n m. C m F(C n ) can be viewed as a rectangle with width (its r-dimension) O(n β 1 β ), height (the ϕ- dimension) O(m 2 ) and weight O(n 1 ). So the measure of C m F(C n ) is given by 1 µ(c m F(C n )) m 2 n 2+ 1 β 1 On the other hand, each cell C m has measure O(m 3 ). Thus there exists constant c, such that m µ(c m F(C n )) µ(c m ) = cm n 2+ 1 β 1 + o(m 1 ) Let W be an unstable curve. Denote by m W the Lebesgue measure on W. For every x W and n 0 denote by W n (x) the smooth component of F n (W) containing the point F n (x) and by (4.6) r n (x) = r Wn(x)(F n x) the distance from the point F n (x) to the nearest endpoint of W n (x). Clearly, r n (x) is a function on W that characterizes the size of smooth components of F n (W). Consider a short unstable curve W M, in [7], the following lemma was proved: 16

17 Lemma 8. If the one-step expansion (3.2) hold, then there exists a constant c 1 > 0, such that for all n 0 and δ > 0 (4.7) m W (r n (x) < δ) c 1 δ Lemma 9. There exist a, e > 0, such that for any large m, there exists C m C m, with µ(c m \ C m ) m a µ(c m ), and any x C m, Fx, F 2 x,..., F b ln m x all belong to cells with index less than m 1 e. Proof. For n satisfies (2.6). Thus for any ε > 0 small, it follows from Lemma 7, n=m µ(f 1 (C n )/C m ) = O(m β 1 β +ε εβ 1 β ) Thus we can neglect points x C m such that F(x) C n with n > m β 1 β +ε and those for which F 2 (x) C k with k > m ( β 1 β )2 +ε. It remains to estimate the probability that points y C k with k m ( β 1 β )2 +ε will come up to C i, i m 1 e, within O(ln m) iterations of F, where e > 0 is to be determined below. To guarantee distortion bound, we only need to deal with homogenous sections of k cells. Every cell C k has length of order k 1/β, and it is divided into homogeneous sections of length j 3 for j k β 1 2β. It also follows that in each cell Ck, for j k β 1 2β, the section F 1 (H j ) C k has height O(j 5β 1 1 β ), We will only keep sections with j k β 1 2β +ε, since the union of the rest has measure 1 1 k Ck 3 c = O(µ(C k )/k c ), β 1 j=k 2β +ε j 5β 1 β 1 where which is negligible with c = 4εβ. β 1 We foliate the j th section F 1 (H j ) C k by short smooth unstable curves, such that their images in other cells will be homogenous unstable curves stretching completely across homogenous section H j (with negligible exceptions). Then we consider an arbitrary unstable curve W C k, k m β 1 2β +ε. It is not hard to see the length of W satisfies W k β 1 β. Denote Wj to be the homogenous portion of W in the j th section, j k β 1 2β +ε. Its length is W j > j 2β 2 2β2 1 (β 1) 2 d ε(1+ k (β 1) 2 ) β 1 d( β m )2 1 β 2, 17

18 where d = β 1 + β. 2β β 1 Next we use Lemma 8, notice that if F n (x) C k, then the length of the largest homogenous unstable manifold in C k is O(k (β 1) 2β β β 1 ). So applying (4.7) with δ = m d(1 e), where e > 0 is small enough. Notice δ W j, so there exists a > 0 such that m Wj (r n < m d(1 e) ) c 1 m d(1 e) Cm a W j. Let I = [n 0, n 1 ] be the longest interval, within [1, m], between successive returns to M. Without loss of generality, we assume that m n 1 n 0, i.e. the leftover interval to the right of I is at least as long as the one to the left of it (because the time reversibility of the billiard dynamics allows us to turn the time backwards). Due to Lemma 9, for large measure of typical points y M I we have F t (y) M mt where m t decreases exponentially fast. So there exists c > 0, such that m/2 I + b I 1 e ln I c I which gives I m. 2c Let D m = {x A m \ B m,b : I m }. Thus it is enough to estimate the 2c size of D m. Since for any x D m, one of its forward images belongs to m I with I m. Applying the bound (4.1) to the interval I gives 2c (4.8) µ(d m ) m m ( m 2c ) 3 = Cm 1 (the extra factors of m must be included because the interval I may appear anywhere within the longer interval [1, m], and the measure µ is invariant). In terms of Young s tower, we obtain (4.9) µ(x : R(x; F, 0 ) > m) Cm 1 m 1 This completes the proof of the theorem. 5 Remarks and future works The family of billiards we study here has never been studied before. So it would be very interesting to compare its statistical properties and dynamical 18

19 properties with those of famous billiards such as (semi-)dispersing billiards whose dynamical properties have been investigated in great detail (see, for example in [2, 3, 6]). These include sharp estimates on distortion bounds, conditional densities on unstable manifolds, the Jacobian of the holonomy map, ergodic and mixing properties, etc. For the class of billiards we study in this paper, we believe all above facts still hold, the proofs should be similar to dispersing billiards with infinite horizon. We plan to publish separately a detailed investigation of those properties for this class of billiards with fla On the other hand, we only study correlation rates for the case when β (2, 1+ 2), since the one-step expansion estimate (5) fails otherwise. So to overcome this difficulty, we might have to improve the general methods developed in [7]. We believe this can be done by getting a much weaker version of Growth Lemma [6]. The work on these questions are currently underway. Notice we can always replace Γ with a dispersing boundary without changing the mixing rates. Since then it may generate another infinite horizon point, which based on a non-flat point. By [9], for this type of singularity the correlations also decay with order O(m 1 ). Acknowledgement. This paper is written in memory of Professor Robert Kauffman. It was him who lead me to the beautiful world of mathematics. I would also like to thank my advisor N. Chernov for many useful discussions and suggestions. The author was partially supported by SRF for ROCS, SEM. 6 Appendix: Proof of Proposition 3 The proof is similar to the proof of Proposition 1 in [8]. For simplicity, we only point out the big changes we made in the proof. For every point x = (r, ϕ) M, let B(x) be defined as in (2.9). Given an unstable manifold W C m,k (or W C m,k ) and a point x W, the map F = F km expands W at x by the factor [3, 5] k 1 ( (6.1) Λ m,k (x) p = 1 + τ(xj )B(x j ) ) where x j = (r j, ϕ j ) = F jm (x). j=0 19

20 The expansion factor (6.1) is measured in the p-norm. But in terms of the expansion factor defined by the Euclidean norm (6.2) v = [ (dr) 2 + (dϕ) 2] 1/2 we have k 1 ( (6.3) Λ m,k (x) 1 + τ(xj )B(x j ) ) cosϕ j cosϕ j+1 j=0 where x j = F jm (x). The initial value B(x), x W, is bounded away from zero and infinity: B min B(x) B max, where B min > 0 is determined by our choice of ε. B(x j ) satisfies the recurrent formula (6.4) B(x j ) = 2K(r j) 1 + cosϕ j τ(x j 1 ) + 1/B(x j 1 ), where (r j, ϕ j ) = x j and τ(x j 1 ) is the distance between x j 1 and x j in the unfolding table Q. Clearly, τ(x j ) 2m, for any j = 1,..., k 1 and it is easy to compute (6.5) K(r j ) β(β 1) r j β 2. Next we consider the trajectory of a point x C m,k (the case x C m,k is easier and will be treated later). For the fixed x, let s x be the distance from x to γ ym in Q. Define a variation flow {ψ(s, t) : s [0, s x ], t (, )} of γ ym such that ψ(0, t) has the same trace as γ ym and ψ(s x, t) has the same trace as γ x. Assume the variation vector field J(t) = ψ (s s x, t) is perpendicular to the trajectory of x, then the above variation flow is unique and J(t) is also called a (generalized) Jacobi field along γ x, see [10]. Denote by J j the corresponding Jacobi vector based at π(x j ). Notice J = dj is also a vector dt field along γ x. Correspondingly, we denote by v j = J j, which also represents the angle made by the F jm (y m ) and x j. Note that (βr β 1 j, 1) is the inward normal vector to Q at the point r j. Geometric considerations (more precisely, the generalized Jacobi equations) yield the following relations: (6.6) v j v j+1 = 2 arctan(βr β 1 j+1 ) J j J j+1 = 2m tanv j + 2m(r β j + rβ j+1 ) tanv j. 20

21 Also notice r j+1 = J j+1 cosϕ j+1. Since C m,k contains points whose images will be trapped in the window U m during the next k 1 iterations of F m. So by (2.8), cosϕ j 1, j = 1, 2,..., k 1. m Thus using Taylor expansion we obtain (6.7) v j v j+1 = 2β(mJ j+1 ) β 1 R v,j+1 J j J j+1 = 2mv j + R J,j, where (6.8) R v,j+1 (mj j+1 ) 3(β 1) > 0 and (6.9) R J,j+1 (mj j ) β (mv j ) > 0 (the positivity of R v,j+1 and R J,j+1 is guaranteed by the smallness of ε and by geometry, v j J β 1 j ). Let k be uniquely defined by J k +1 J k. First we consider the interval 1 j k, i.e. where {J j } is decreasing. Note that both {J j } and {v j } are decreasing sequences of positive numbers for j = 1,...,k. Lemma 10. Let k [1, k ] be uniquely defined by the condition (6.10) v k 1 > 2v k > v k. Then for all 1 < j k we have (6.11) 2(mJ j ) β (mv j ) 2 (mj) 2β 2 β. More precisely, (6.12) (mj j ) 2 β 2 2(β 2)(mj) + C 1 ln(mj) + C 2 mj ( j k ) 2β β 2. Furthermore, (6.13) k k and mj k (mk ) 2 2 β 21

22 The proof of this lemma follows from similar arguments as in the proof of Lemma 3-Lemma 6 in Appendix of [8]. In fact we can get an idea to understand the above lemma from the following estimations. (6.7) can be approximated by the following equations: (6.14) J i J j+1 = 2β(mJ j+1 ) β 1, J j J j+1 = 2mJ j. For j large enough, the solutions of the above systems can be approximated by the solutions of the following differential equation: (6.15) J(t) = βm β 2 J β 1 (t). Assume J(t) = AJ a (t), and plug into the above equation, we get A = 2m β 2 and a = β/2. Thus the solution of (6.15) is (mj(t)) 1 β/2 = 2(1 β/2)t + (mj(0)) 1 β/2 By the second equation in (6.14) and the definition of U m, (mj j ) 1 β/2 2(1 β/2)2mj + (mj(0)) 1 β/2 2(2 β)mj + O( m). Corollary 11. For all 1 j k we have (6.16) ] 2 2K(r j ) D [mj + C 1 cosϕ ln(mj) j where D = mβ(β 1) (β 2) 2. Lemma 12. For all 1 j < k we have (6.17) B(x j 1 ) A mj + C 1 ln(mj) (mj) 2, where A > 0 satisfies 2mA 2 ma = mβ(β 1) (β 2) 2, hence A = β 1 β 2. Now we are ready to estimate the expansion factor Λ m (x) given by (6.1). Lemma 13. (6.18) k 1 j=1 where C > 0 is a constant. ( 1 + τ(xj )B(x j ) ) Ck 2(β 1) β 2 22

23 Proof. Note that τ(x j ) > 2m. Hence, due (6.17), we have ln [ k 1 ( 1 + τ(xj )B(x j ) )] k [ 2mA > mj + 2C ] 3 ln(mj) (mj) 2 j=1 j=1 with some large constant C 3 > 0. Therefore, ln [ k 1 ( 1 + τ(xj )B(x j ) )] > 2A lnk + const > 2A lnk + const, j=0 where the last inequality follows from (6.13). Lastly, note that 2A = 2(β 1) β 2, which completes the proof of the lemma. By (2.13) 1 + τ(x 0 )B(x 0 ) 1 + τ(x 0)K(r 0 ) cosϕ 0 cm 1 1 β The bound (6.18) implies the expansion factor in p norm is (6.19) Λ (1) m,k (x) p : = k 1 j=0 ( 1 + τ(xj )B(x j ) ) Cm 1 1 β k 2(β 1) β 2 Notice for j [1, k ], cosϕ j 1/m, so the corresponding expansion factor in p norm is equivalence to the factor in Euclidian norm. Also notice cosϕ 0 cm 1 β 1, thus the total expansion factor in Euclidian norm is (6.20) Λ (1) m,k (x) Lemma 14. k 1 j=0 ( 1 + τ(xj )B(x j ) ) cos ϕ j Cmk 2(β 1) β 2 cos ϕ j+1 k 1 (6.21) Λ (2) ( m,k (x) : = 1 + τ(xj )B(x j ) ) Cm 1 β β k β 2 j=k where C > 0 is a constant. 23

24 Proof. We will use the time-reversibility of the billiard dynamics as in [9]. Let V u and V s be two unit vectors tangent to the unstable and stable manifolds, respectively, at the point x. It follows from (6.4) that (6.22) B(x 0 ) 2 K(r 0) cosϕ 0. Now (2.9) implies that the slope of the vector V u k is dϕ dr = cos ϕ 0 B(x 0 ) K(r 0 ). By (2.12), we have dϕ dr > C m for some constant C > 0. Hence the vector V u makes an angle greater than C with the horizontal r-axis. By the time reversibility, the vector V s makes m an angle less than C with the horizontal r-axis. Thus the area of the m parallelogram Π spanned by V u and V s is of order 1/m. Consider the parallelogram Π k = D x F k m (Π) spanned by the vectors Vk u = D xf k m (V u ) and V s k = D xf k m (V s ). Since the map F k m preserves the measure dµ = cos ϕ dr dϕ, we have cosϕ k Area(Π k ) = cosϕ 0 Area(Π). Note that cosϕ 0 cm 1 β 1 and cos ϕ k m 1, hence On the other hand, Area(Π k ) Area(Π) cosϕ 0 cosϕ k cm 1 β 1. Area(Π k ) = Vk u s Vk sin γ k where Vk u s and Vk denote the lengths of these vectors in the Euclidean norm (6.2) and γ k denotes the angle between them. Next we estimate γ k. It easily follows from (6.17) that (6.23) B(x k ) c 2 1 mk. 24

25 Now (2.9) and (6.4) implies that dϕ dr > C m 2 k for some constant C > 0. Hence sin γ k > c 1 /(m 2 k) for some constant c 1 > 0, and we obtain Vk u s Vk < c 2 m 2+ 1 β k for some constant c 2 > 0. Obviously, Vk u = Λ(1) m,k (x) V u Λ (1) m,k (x). By the time reversibility of the billiard dynamics, the contraction of stable vectors during the time interval (0, k ) is the same as the expansion of the corresponding unstable vectors during the time interval (k, k), hence Therefore, Vk s Λ(2) m,k (x) 1 V s Λ (2) m,k (x) 1. (6.24) Λ (2) m,k (x) > Λ(1) m,k (x) ckm 1+ 1 β for some constant c > 0. Now (6.24) and (6.20) imply (6.21). Combining (6.20) and (6.21) gives Λ m,k (x) Λ (1) m,k (x) Λ (2) m,k (x) Cm 1 1 β k 4(β 1) β 2 1. This proves Proposition 3 for W C m,k. We now consider the remaining case W C m,k. In that case k can be defined as the turning point, i.e. by J k < J k 1 and J k < J k +1. Observe that if x = (r, ϕ ) C m,k, then there exists another point x = (r, ϕ) C m,k with r = r and ϕ < ϕ, whose trajectory goes through the window. Since ϕ < ϕ, it follows that the r-coordinate of the point F jm (x) will be always smaller than the r-coordinate of the point F jm (x ), for all 1 j k. This observation and the bound (6.12) that we have proved for J j implies that the same bound holds for J k and for all 1 j k. The rest of the proof of Proposition 3 for x C m,k is identical to that of the case x C m,k. Proposition 3 is now proven. 25

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