Decay of correlations for billiards with flat points I: channel effect
aa r X i v : . [ m a t h . D S ] M a y Decay of correlations for billiardswith flat points I: channel effect
Hong-Kun Zhang September 14, 2018
Abstract
In this paper we constructed a special family of semidispersing billiards boundedon a rectangle with a few dispersing scatterers. We assume there exists a pair of flatpoints (with zero curvature) on the boundary of these scatterers, whose tangent linesform a channel in the billiard table that is perpendicular to the vertical sides of therectangle. The billiard can be induced to a Lorenz gas with infinite horizon whenreplacing the rectangle by a torus. We study the mixing rates of the one-parameterfamily of the semi-dispersing billiards and the Lorenz gas on a torus; and show thatthe correlation functions of both maps decay polynomially.
A billiard is a mechanical system in which a point particle moves in a compact container Q and bounces off its boundary ∂Q . It preserves a uniform measure on its phase space, and thecorresponding collision map (generated by the collisions of the particle with ∂Q , see below)preserves a natural (and often unique) absolutely continuous measure on the collision space.The dynamical behavior of a billiard is determined by the shape of the boundary ∂Q , andit may vary greatly from completely regular (integrable) to strongly chaotic.In this paper we consider a planar semidispersing billiard table Q on a rectangle R .There are a few convex obstacles B = ∪ i B i inside R , such that the billiard table is definedas Q = R \ B . We assume that the boundaries of these obstacles have positive curvaturesexcept on two opposite points in the table, p and p , which we call the flat points, asthey have zero curvature; we also assume that the tangent lines of these two flat pointsare parallel and perpendicular to a pair of straight boundaries of R . When replacing the Department of Mathematics & Statistics, University of Massachusetts Amherst, MA 01003; Email:[email protected] R by the torus, this becomes a classical Lorenz gas with infinite horizon, and wecan say that there is an infinite channel in the unfolding table that bounded by tangentlines of these two flat points.The billiard flow Φ t is defined on the unit sphere bundle Q × S and preserves theLiouville measure. Both Lorentz gas and the semi-dispersing billiards have been proven toenjoy strong ergodic properties: their continuous time dynamics and the billiard ball mapsare both completely hyperbolic, ergodic, K-mixing and Bernoulli, see [3, 16, 18, 19, 20] andthe references therein. However, these systems have quite different statistical propertiesdepending on the geometric properties of the billiard table.When all scatterers have positive curvature, the Lorentz gases were proven to have fastmixing rates. Exponential mixing rates were obtained by Young [22] for finite horizon caseand by Chernov [4] under the condition of infinite horizon. On the other hand the semi-dispersing billiards have much weaker statistical properties. Based upon the methods ofYoung [22], it was proven that the mixing rates are of order O (1 /n ), as n → ∞ , see [10, 12].See also the recent papers by Chernov, Dolgopyat, Szasz and Varju [6, 14] and the referencestherein for related studies on other statistical properties of these systems.In the current paper we relax the assumption of strictly positive curvature on the bound-ary of dispersing scatterers, by adding finitely many flat points (with zero curvature) on theboundary of dispersing scatterers. A family of dispersing billiards with flat points (as a per-turbation of Sinai billiards with finite horizon) were constructed by Chernov and Zhang [11].It was proved that the mixing rate varies between O (1 /n ) and exponentially fast dependingon the parameter.Here we consider modifications of the billiards considered in [11], by consider a per-turbation of semidispersing billiards on a rectangle, whose induced map can be viewed asa perturbation for Sinai billiards with infinite horizon. If we replace the rectangle by thetorus, and view the billiard system in the unbound table with periodic scatterers, thus theassociated Lorentz gas has an infinite channel bounded by the tangent lines of these twoflat points.To simplify our analysis, we assume there is only one convex scatterer B located at thecenter of the table, and assume there are 4 flat points on the boundary ∂ B . In addition, ∂ B is symmetric about the vertical and horizontal lines passing through the center of ∂ B .Thus the geometric feature of these special points are essentially identical. We fix any β ∈ (2 , ∞ ). Denote p as one of the flat point and let { ( s, z ) } be the Cartesian coordinatesystem originated at p , then the part of the boundary ∂ B that containing p can be locallyviewed as the graph of z = z ( s ), such that for some small ε > h1 ) z ( s ) = −| s | β + O ( | s | β +1 ), for any | s | < ε ;( h2 ) The tangent line at p is parallel to one straight side of the rectangle R .The first assumption ( h1 ) implies that these special points are indeed flat points (with zero2urvature) for β >
2. In addition ( h2 ) implies that any trajectory with infinite horizon isonly tangent to the scatterer at these special points. Fig.1 (a) describes a billiard table withflat points for β >
2, and x p is a vector whose trajectory has infinite horizon.PSfrag replacements RR pp x p x p B B
Fig. 1: (a). Billiards on a rectangle with 4 flat points; (b). Billiards on a torus with 4 flat points.
There is a natural cross section M in Q × S that contains all postcollision vectors basedat the boundary of the table ∂Q . The set M = ∂Q × [ − π/ , π/
2] is called the collisionspace. Any postcollision vector x ∈ M can be represented by x = ( r, ϕ ), where r is thesigned arclength parameter along ∂Q , and ϕ ∈ [ − π/ , π/
2] is the angle that x makes withthe inward unit normal vector to the boundary. The corresponding Poincar´e map (or thebilliard map) F : M → M generated by the collisions of the particle with ∂Q preserves anatural absolutely continuous measure µ on the collision space M .For any square-integrable observable f, g ∈ L µ ( M ), correlations of f and g are definedby(1.1) C n ( f, g, F , µ ) = Z M ( f ◦ F n ) g dµ − Z M f dµ Z M g dµ It is well known that a system ( F , µ ) is mixing if and only if(1.2) lim n →∞ C n ( f, g, F , µ ) = 0 , ∀ f, g ∈ L µ ( M ) . As a result, the rate of mixing of ( F , µ ) 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 bounded,piecewise H¨older continuous or bounded piecewise H¨older continuous with singularities co-incide with those of the map F . We denote W u/s as the collection of all unstable/stablemanifolds for the billiard map.For any γ >
0, let H − ( γ ) be the set of all bounded real-valued functions f ∈ L ∞ ( M , µ )such that for any x and y lying on one stable manifold W s ∈ W s ,(1.3) | f ( x ) − f ( y ) | ≤ k f k − γ dist( x, y ) γ , with k f k − γ : = sup W s ∈ W s sup x,y ∈ W s | f ( x ) − f ( y ) | dist( x, y ) γ < ∞ . x, y ) denotes the Euclidian distance of x and y in M , Similarly, we define H + ( γ )as the set of all bounded, real-valued functions g ∈ L ∞ ( M , µ ) such that for any x and y lying on one unstable manifold W u ∈ W u ,(1.4) | g ( x ) − g ( y ) | ≤ k g k + γ dist( x, y ) γ , with k g k + γ : = sup W u ∈ W u sup x,y ∈ W u | g ( x ) − g ( y ) | dist( x, y ) γ < ∞ . For every f ∈ H ± ( γ ) we define(1.5) k f k ± C γ : = k f k ∞ + k f k ± γ . In this paper we obtain the following results.
Theorem 1.
For the family of semidispersing billiards on a rectangle defined as in Fig. 1(a), if β > , then the correlations (1.1) for the billiard map F : M → M and piecewiseH¨older continuous functions f ∈ H − ( γ ) , g ∈ H + ( γ ) , decay polynomially: (1.6) | C n ( f, g, F , µ ) | ≤ C k f k − C γ k g k + C γ · n − η where η = 1 . Theorem 2.
For the family of Lorenz gas on a torus defined as in Fig. 1 (b), if β > ,then the correlations (1.1) for the billiard map ˜ F : ˜ M → ˜ M and f ∈ H − ( γ ) , g ∈ H + ( γ ) ,decay polynomially: (1.7) | C n ( f, g, ˜ F , ˜ µ ) | ≤ C k f k − C γ k g k + C γ · n − η where η = β +2 β − , and ˜ µ is the conditional of µ on ˜ M . Moreover, we also have the optimal bound, as obtained by [7].
Theorem 3.
For both billiard systems as described above, consider any function g with g ∈ H + and supp( g ) ⊂ M . Then there exists N = N ( g ) > , such that for any n > N , andany f ∈ H − supported on M , the correlations (1.1) decay as: (1.8) C n ( f, g ) = µ ( R > n ) µ ( f ) µ ( g ) + k f k − C γ k g k + C γ o ( n − η ) . Convention. We use the following notation: A ∼ B means that C − ≤ A/B ≤ C for someconstant C > A = O ( B ) means that | A | /B < C for some constant C >
0. Fromnow on, we will denote by
C >
General scheme
Based upon the methods by Young [22] and Markarian [17], a general scheme was developedin [10, 11, 12, 13] on obtaining slow rates of hyperbolic systems with singularities and appliedthe method on different models. In the general scheme, first one needs to ‘localize’ spotsin the phase space where expansion (contraction) of tangent vectors slows down. Let ¯ M denote the union of all such spots and M = M \ ¯ M . One needs to verify that the return map F : M → M (that avoids all the ‘bad’ spots) is strongly (uniformly) hyperbolic. One caneasily check that it preserves the measure µ M obtained by conditioning µ on M . For any x ∈ M we call R ( x ) = min { n ≥ F n ( x ) ∈ M } the return time function and the returnmap F : M → M is defined by(2.1) F ( x ) = F R ( x ) ( x ) , ∀ x ∈ M In order to prove the main Theorems, the the strategy developed by Chernov and Zhangconsists of three steps; they are fully described in [10, 12] (as well as applied to severalclasses of billiards with slow mixing rates), so we will not bring up unnecessary details here. (F1)
First, the map F : M → M enjoys exponential decay of correlations. More precisely, for any piecewise Holder observables f, g on M , with f ∈ H − ( γ ) , g ∈ H + ( γ ), for γ ∈ (0 , Z M ( f ◦ F n ) g dµ M − Z M f dµ M Z M g dµ M ≤ C k f k − C γ k g k + C γ ϑ n for some uniform constant ϑ = ϑ ( γ ) ∈ (0 ,
1) and
C > (F2) Second, the distribution of the return time function R : M → [1 , ∞ ) satisfies: µ (( x ∈ M : R ( x ) ≥ n )) ∼ n η , where η > n ≥ It was shown in [10] that the same order of upper bound as in Theorem 1 follows from ( F1 )-( F2 ).Also, the proof of ( F1 ) is reduced in [10] to the verification of a one-step expansion condition. The following lemma was proved in [10].
Lemma 4.
For systems under assumption (
F1-F2 ), for the billiard map F : M → M andany piecewise H¨older continuous functions f ∈ H − ( γ ) , g ∈ H + ( γ ) on M , the correlations(1.1) decay as (2.2) | C n ( f, g, F , µ ) | ≤ C k f k − C γ k g k + C γ n − η (ln n ) η for some constant C > . F1 ) - ( F2 ) for η = 1(or η = 1 + β − , respectively), except the extra logarithmic factor. To improve the upperbound for the decay rates, one needs to analyze the statistical properties of the returntime function. In [12, 7], the upper bound of decay rates of correlations was improved bydropping the logarithmic factor.The paper is organized as following. In Section 3, we derive properties for the intermedi-ate map – the Lorenz gas on a torus. We also construct an induced system ( F, M ) for bothmaps, by removing collisions on rectangular boundary as well as a small neighborhood ofthe flat points. The hyperbolicity of the reduced map is proved in Section 4. The assump-tion (F2) is verified in Section 5, by analyzing the distribution of the return time function.Details of the singular sets is given in Section 6. The regularity of unstable curves is studiedin Section 7, including the property of the bounded curvature, distortion bounds, as well asthe absolute continuity of the stable holonomy maps. The exponential decay of correlationsfor the reduced map and (F1) is proved in Section 8, by verifying the One-step expansionestimates, see Lemma 13. In Section 9, we prove the improved upper bound using methodas in [12]. ( F, M ) ( ˜ F , ˜ M ) . In order to construct a good set in the phase space M , we first consider an intermediatesystem – the associated Lorentz gas on torus. More precisely, we replace the rectangle R by a torus T , then we have an unbounded table ˜ Q , which is obtained by unfolding T \ B .We denote the corresponding system as ( ˜ F , ˜ M ), with˜ M = { x = ( r, ϕ ) ∈ M | r ∈ ∂ B } , ˜ F = F | ˜ M Note that the system ( ˜
F , ˜ M ) counts only collisions on the convex scatterer B . ˜ S := ∂ ˜ M ∪ ˜ F − ∂ ˜ M is the singular set of ˜ F . Since the boundary ∂Q is C , the map˜ F : ˜ M \ ˜ S → ˜ M \ ˜ F ˜ S is a local C diffeomorphism. For any x ∈ ˜ M , let τ ( x ) be the length of the free pathbetween the base points of x and that of the next non-tangential postcollision vector in theunbounded region ˜ Q . A point x ∈ ˜ M is said to be an IH singular point if its free path isunbounded in ˜ Q , i.e. its forward trajectory never experiences any non–tangential collisionswith the boundary of the scatterer (as if the flat sides of R are transparent). In particularwe denote by x p an IH points based at the flat point p . Then there exists a channel orcorridor in ˜ Q that contains the trajectory of x p . We assume the r -coordinate of p is 0, then6 p = (0 , π/ ∈ ˜ M . By the symmetric property of the billiard table, it is enough to considerthose components in the vicinity of the IH point x p .Let x = ( r, ϕ ) and ˜ F x = ( r , ϕ ). According to [8], for any x ∈ ˜ M \ ˜ F − ∂ ˜ M thedifferential of ˜ F is D x ˜ F = − ϕ (cid:18) τ K ( r ) + cos ϕ ττ K ( r ) K ( r ) + K ( r ) cos ϕ + K ( r ) cos ϕ τ K ( r ) + cos ϕ (cid:19) (3.1)where τ = τ ( x ) is the distance between the base points of x and ˜ F ( x ) in the unboundedtable ˜ Q . In addition, for x ∈ ˜ M ∩ ˜ F − ∂ ˜ M , ˜ F x is tangential, we define τ ( x ) as the distancebetween the base of x and the first non-tangential collision along the forward trajectory of x . By assumption the boundary of B is C smooth, so the map ˜ F is C smooth on eachsmooth component of ˜ M \ ˜ S . The dynamical properties of the intermediate system ( ˜ F , ˜ M )was studied in [23]. It was shown in [23] that the singularity of the free path τ divides theregion ˜ M near x into countably many connected components, labeled as M n := ( x ∈ ˜ M : τ ( x ) = n )for n ≥
1. It was proved for n ≥
1, an n -cell M n is the domain bounded by some smoothcurves s n , s n +1 , s ′ and ϕ = π/
2, such that τ is smooth on M n . Then for any x ∈ M n , thefree path τ ( x ) is approximately of length n units. To investigate statistical properties of thebilliard map, it is crucial to characterize the distribution of the free path τ . The followinglemma was proved in [23]. Lemma 5.
For β ∈ [2 , ∞ ) . Let x p = (0 , π/ be the IH point based at p .(1) Let s ⊂ ˜ F − ∂ ˜ M be any smooth curve with equation ϕ = ϕ ( r ) in the vicinity of x p .Then s is a C decreasing curve with slope: dϕ/dr = − cos ϕτ − K .(2) The curve s ′ satisfies (3.2) π − ϕ = βr β − + r β + O ( r β +1 ) , ∀ r ∈ [0 , ε ] (3) For n large, the curve s n is stretched between ϕ = π/ and s ′ , with equation satisfying (3.3) π − ϕ = βr | r | β − + 1 n + O ( n − β +1 β − ) , ∀ r ∈ [ − ε, ε ] (4) M n has length (or r -dimension) of order O ( n − β ) , width (or ϕ -dimension ) of order O ( n − ) and µ -density of order O ( n − ββ ) . In addition all boundary components of M n have uniformly bounded curvature.
5) There exist c > c > that do not depend on β such that (3.4) c n − + O ( n − − β ) ≤ µ ( M n ) ≤ c n − + O ( n − − β ) (6) There exist positive constants c < c , such that for any n ≥ , if M m ∩ ˜ F M n = ∅ then (3.5) c β √ n β − ≤ m ≤ c β − √ n β Although the intermediate system omit all collisions on the boundary of the rectangle R , it still has points with zero Lyapunov exponents. It was shown in [23] that if β >
2, theintermediate system ( ˜
F , ˜ M ) is nonuniformly hyperbolic. Let(3.6) A = { x = ( r, ϕ ) ∈ M : K ( x ) = 0 , K ( ˜ F x ) = 0 } . The following facts were also proved in [23]:
Lemma 6.
For any β ∈ (2 , ∞ ) , A is a null set on which the Lyapunov exponent is zero.Moreover A is not empty and any x ∈ ˜ M \ A has nonzero Lyapunov exponent. One can check that A is made of periodic points based at those flat points. Indeed forany n ≥
1, there exists a unique periodic point y n ∈ A ∩ M n . We define its stable set as w sn ( y n ) := { x ∈ M n | lim m →∞ d ( ˜ F m x, y n ) = 0 } . Since the Lyapunov exponents are zero at points in A , we need to eliminate points in A and the union of their stable sets, denoted as w s . ( F , M ) and ( ˜ F , ˜ M ) . In this subsection, we will construct an induced map (
F, M ) that works for both systems( F , M ) and ˜ F , ˜ M ). Later on, we will investigate the induced map ( F, M ), and to investigateboth maps ( F , ˜ M ) and ( F , M ) simultaneously.Note that by Lemma 5 and [23], the zero curvature line r = 0 cut a cell M m in to twoparts, one of which has very small measure. We fix a small constant ε >
0, and for any m ≥
1, we define a set U m = ([ − ε m − β − , ε m − β − ] × [ − π/ , π/ ∩ M m . Here ε can be chosen small enough such that M m \ U m still contains two regions with r -dimension much larger than that of U m . Let(3.7) M = { x = ( r, ϕ ) ∈ ˜ M : r ∈ ∂ B , x ∈ M m \ ( U m ∪ w s ) , m ∈ N } ) , M only contains points that based on the convex scatterer B , and we also removefrom each cell M m the stable set w sm and a narrow window U m that contains y m , see Figure. 2.Let q m denote one of the two points on ∂Q that border the base of U m , and by q m the otherbase point.First of all we define a return time function related to the special set M , such that forany x ∈ M ,(3.8) R ( x ) = min { n > F n ( x ) ∈ M } We define the map F : M → M as the first return map to M , such that for any x ∈ M , F x = F R ( x ) x We call C m := ( x ∈ M : τ ( x ) = m ) = M m \ U m = C ′ m ∪ C ′′ m as the m -cell of F in M (note this is not the level sets of R ). C m contains two disconnectedregions from M m , denoted as C ′ m and C ′′ m , where C ′ m contains almost tangential collisionswith angle smaller than 1 /m , while C ′′ m contains collisions with angle larger than 1 /m .Moreover, the set U m has length ( r -dimension) O ( m − β − ), height ( ϕ -dimension) O ( m − )and density O ( m − ), thus the measure µ ( U m ) ∼ m − − β − . By Proposition 5, we know that µ ( M m ) ∼ m − . Thus(3.9) µ ( C m ) = µ ( M m ) − µ ( U m ) ∼ µ ( M m ) ∼ m − Notice for any point x ∈ U m , and x = F x , we have(3.10) | r | β − ≤ cm , and τ ( x ) ∼ m. On the other hand let K m be the curvature at the boundary of U m , then for any x = ( r, ϕ ) ∈ C m , the curvature at r satisfies(3.11) Cm − β ≥ K ( r ) = β ( β − r β − ≥ K m ≥ cm − β − Clearly F preserves the conditional measure ˆ µ obtained by restricting µ on M . Next wewill check the reduced system for the three conditions ( F1 ) -( F3 ) proposed in the generalscheme. 9 Hyperbolicity of ( F, M ) The key to understanding chaotic billiards lies in the study of infinitesimal families oftrajectories. The basic notion is that of a wave front along a billiard trajectory. Moreprecisely, for any x ∈ M , let V ∈ T x M be a tangent vector. For ε > γ = γ ( s ) ⊂ M , where s ∈ ( − ε, ε ) is a parameter, such that γ (0) = x and dds γ (0) = V The trajectories of the points y ∈ γ , after leaving M , make abundle of directed lines in Q . To measure the expanding or contracting of the wave front,let B = B ( x ), which represents the curvature of the orthogonal cross-section of that wavefront at the point x with respect to the vector V . We say the wave front γ is dispersing if B >
0. Similarly, the past trajectories of the points y ∈ γ (before arriving at M ) makea bundle of directed lines in Q whose curvature right before the collision with ∂Q at x isdenoted by B − = B − ( x ). We define x = ˜ F x = ( r , ϕ ), then we have [8]:(4.1) B − ( x ) = B ( x )1 + τ ( x ) B ( x ) = 1 τ ( x ) + B ( x ) and B ( x ) = B − ( x ) + R ( x )where R ( x ) = K cos ϕ is called the collision parameter at x . (4.1) implies that a wavefront thatis initially dispersing will stay dispersing. By our construction on the included map and(3.11), for any x ∈ C m , with m ≥
1, there exist τ min > < K m < K max such that forany x ∈ M ,(4.2) τ ( x ) ≥ τ min and K m ≤ K ( x ) ≤ K max . where K m ∼ m − β − is the minimal curvature at the end point of U m . Denote by V = dϕ/dr the slope of the tangent line of W at x . Then V satisfies(4.3) V = B − cos ϕ + K ( r ) = B cos ϕ − K ( r )We use the cone method developed by Wojtkowski [21] for establishing hyperbolicity forphase points x = ( r, ϕ ) ∈ M . In particular we study stable and unstable wave front. Therelations in (4.1) and (4.1) implies that a dispersing wave front remains bounded away fromzero. Definition 7.
The unstable cone C ux contains all tangent vectors based at x whose imagesgenerate dispersing wave fronts: C ux = { ( dr, dϕ ) ∈ T x M : K ( r ) ≤ dϕ/dr ≤ K ( r ) + τ − } Similarly the stable cones are defined as C sx = { ( dr, dϕ ) ∈ T x M : − K ( r ) ≥ dϕ/dr ≥ − K ( r ) − τ − } e say that a smooth curve W ⊂ M is an unstable (stable) curve for the reduced system ( F, M ) if at every point x ∈ W the tangent line T x W belongs in the unstable (stable) cone C ux ( C sx ). Furthermore, a curve W ⊂ M is an unstable (resp. stable) manifold for thereduced system ( F, M ) if F − n ( W ) is an unstable (resp. stable) curve for all n ≥ (resp. ≤ ). For any m ≥
1, consider a short unstable curve W ⊂ C m with equation ϕ = ϕ ( r ). Let x = ( q, v ) = ( r, ϕ ) ∈ W and V = ( dr, dϕ ) be a tangent vector at x of W . It follows from(4.1) that(4.4) B − ( x ) ≤ τ − , K m ≤ cos ϕ B ( x ) < τ − + 2 K max Combining with (3.1), if ˜
F x stays away from the flat points, then the tangent vector of
F W at x = ( r , ϕ ) = ˜ F x satisfies(4.5) K ( r ) + cos ϕ τ ( x ) + cos ϕ K ( r ) ≤ V ( x ) ≤ K ( r ) + cos ϕ τ ( x )This implies that the unstable cone C ux is strictly invariant under ˜ F , if the base of ˜ F x staysaway from the flat points. Similarly one can check that D ˜ F ( C sx ) ⊃ C s ˜ F x . Lemma 8.
Let m be a large number. For any m > m , any unstable curve W ⊂ C m suchthat ˜ F − W is also unstable, let x = ( r, ϕ ) ∈ W .(a) B − ( x ) ∼ τ ( x ) − , B ( x ) ∼ K cos ϕ ≥ cm β where c = c ( Q ) is a constant.(b) The slope V ( x ) = dϕ/dr satisfies V ∼ K ∼ cos ϕ β − β − ≥ K m .Proof. Since m ≥ m is large, the condition ˜ F − W is unstable implies that W is almostparallel to the long boundary of ˜ F C n , for some n ∈ [ m β − β , m ββ − ]. Note that ˜ F C n hastwo part: the smaller region containing points after almost tangential collisions, denoted as˜ F C ′ n ; and the larger region with collision angles smaller than 1 /n , denoted as ˜ F C ′′ n . Thusby (3.2), point x = ( r, ϕ ) ∈ W ∩ ˜ F C ′ n satisfies(4.6) cos ϕ ∼ − β | r | β − + 1 n ∼ n − ∼ K β − , n ∈ [ m β − β , m ]where we have used the definition of ˜ F C ′ n , i.e. | r | ∼ n − β − . Similarly, for x = ( r, ϕ ) ∈ W ∩ F C ′′ n (4.7) cos ϕ ∼ β | r | β − + 1 n ∼ | r | β − ∼ K β − , n ∈ [ m, m ββ − ]11oreover this implies that(4.8) K cos ϕ ∼ ϕ β − ≥ Cm β Thus the slope of the tangent line of W at x satisfies(4.9) dϕ/dr ∼ K + cos ϕτ ∼ K ∼ cos ϕ β − β − ≥ K n Combining with (4.3) and (4.1), we also obtained the following estimates for B ± :(4.10) cos ϕ B ( x ) ∼ K , and B − ( x ) ∼ τ Next we will introduce two metrics on the tangent space T M . Let x ∈ C m , for any m ≥ p-metric on vectors dx = ( dr, dϕ ) by(4.11) | dx | p = cos ϕ | dr | . Put x = ( r , ϕ ) = ˜ F x and dx = ( dr , dϕ ) = D ˜ F ( dx ), for n ≥
1. The expansionfactor is(4.12) | dx | p | dx | p = 1 + τ ( x ) B ( x ) ≥ τ ( x ) K ( r )cos ϕ where τ ( x ) ≥ cm is the collision time for x under ˜ F .(II) Now consider the expansion factor in Euclidean metric | dv | = ( dr ) + ( dϕ ) . Notethat(4.13) | dx || dx | = | dx | p | dx | p cos ϕ cos ϕ q dϕ dr ) q dϕdr ) R It follows from the definition of M , it contains countably many cells { C m } whose boundariesare made of singular curves in ˜ S for the intermediate system ˜ F . On the other hand, thereare new types of singularity curves of F in each m − cell C m consists of preimages of thevertical boundary components of U m ’s. 12or any x ∈ C m , the value R ( x ) corresponding to the number of bounces the billiardtrajectory of the point x ∈ C m reflected by F in the window U m before returning to M . Thesingularities of R ( x ) occur at points where the number of bounces in the window changesfrom k to k + 1 or k −
1, for k >
1. Accordingly, the new singular curves in M correspondingto the discontinuities of the return function R ( x ).We denote { y m } as the sequence of periodic points with period m that approaching tothe IH singular point x p , as m → ∞ . Let w sm = w sm ( y m ) be the weak stable manifolds ofthese periodic points. For each m ≥
1, the singularity set of the map F in C m consists oftwo types of infinite sequences of singularity curves { s m,k } and { s ′ m,k } , approaching to w sm from both sides. These curves correspond to the discontinuities of the function R ( x ) in C m .More precisely, the region above w sm but below c m consists of points whose trajectories enterthe window and turn back without reaching γ y m . This region is divided into a sequence ofalmost parallel strips by { s m,k } , k ∈ N , see Figure. 2. Denote by C m,k the strip bounded by s m,k , s m,k +1 and ∂U m . The region blow w sm but above c ′ m consists of points whose trajectoriesenter the window and manage to move through it crossing γ y m . This region is divided into asequence 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 and ∂U m . Both C m,k and C ′ m,k consist of points experiencing exactly k collisions withthe boundary of Q before returning to M .We also denote C m, = C ′ m, as those points x ∈ C m , such that ˜ F x ∈ M , i.e. these pointsdo not enter into the windows { U m .m ≥ } under iteration of ˜ F . Moreover, for any n > R can be represented as(5.1) ( x ∈ M : R ( x ) = n ) = [ m ≥ [ k =[ n/m ] ( C ′ m,k ∪ C m,k ) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx PSfrag replacements M s ′ m,k U m U m +1 y c ′ m s ′ m,k c m z S C m w sm S m +1 ϕr S m Figure. 2: Singularity curves of F in the vicinity of C m .Fig 2 shows the structure of these new singular curves in the cell C m . There are boldsteeply decreasing curves c m terminating on s or ∂M , consist of the points in C m whose13rajectories hit the point q m of ∂Q under the map F . The stable manifold w sm crosses c m ,consists of points whose trajectories converge to γ y m . The dashed part of w sm does not enterthe window immediately, but will do so in one iteration.In order to determine the rates of the decay of correlations we need certain quantitativeestimates on the measure of the regions C m,k and C ′ m,k and on the factor of expansion ofunstable manifolds W ⊂ C m,k and W ⊂ C ′ m,k under the map F . We first state a lemmaabout the expansion of points in C m,k along all collisions in the region U m . Proposition 9.
For any m ≥ , k ≥ , let W ⊂ C m,k be an unstable curve, and Λ( x ) be itsexpansion factor under the map F = F mk at x ∈ W .(1) The expansion factor in the Euclidian metric satisfies: Λ( x ) ∼ m K ( x ) k β − ∼ m β − k β − (2) If W is a vertical curve then Λ( x ) ∼ m k β − . Here C > is constant. The proof of this proposition is rather lengthy, and will be given in the Appendix usinga similar approach as in [11].
Lemma 10.
For any m ≥ , k ≥ , the set C m,k have measure of order O ( m − − β − k − − β − ) .Proof. Due to the time-reversibility of the billiard dynamics, the singular curves in
F M havea similar structure. Furthermore, the short sides of C m,k stretch completely in U m under˜ F m ( k − , while under F mk they are transformed into long sides of F ( C m,k ), with length ∼ m − β . Let h m,k denote the length of a vertical curve W that stretch completely in C m,k .Thus by Proposition 9, the expansion factor on W is of order O ( m k β − ). This impliesthat(5.2) h m,k ∼ | F W | m k β − ∼ m β − k β − . It follows from Lemma 5 that the length of C m,k is O ( m − /β ), and the density on C m,k isof order O ( m − β ). According to (5.2), the measures of both C m,k and C ′ m,k are of order O ( m
3+ 1 β − k
3+ 4 β − ).It follows from above estimation, we know that ∪ k ≥ µ ( C m,k ) ∼ m − − β − ∼ µ ( U m )Note that from (3.9), we have µ ( C m ) ∼ m − . This implies that in this sequence { C m,k .k ≥ } , only the first set C m, dominates, as(5.3) µ ( C m, ) ∼ µ ( C m ) ∼ m − n ≥
1, we know that the set( x ∈ M : R ( x ) ≥ n ) ⊂ ( ∪ ∞ m = n C m ) [ ∪ n − m =1 X k ≥ n/m C m,k Note that n − X m =1 X k ≥ n/m µ ( C m,k ) ∼ n − X m =1 X k ≥ n/m m β − k β − ∼ n β − (5.4)According to (5.3) we know that µ ( C m, ) ∼ µ ( R = m ) ∼ m − It is immediate that for any large n , the set C n, not only dominates in ( R = n ). but alsodominates the set M n = ( x ∈ ˜ M : τ ( x ) = n ). This implies that µ ( x ∈ M : R ( x ) ≥ n ) ∼ µ ( x ∈ ˜ M : τ ( x ) ≥ n ) ∼ n . (5.5)This verifies condition ( F2 ) with η = 1 for the system ( F , M ).Moreover, for the intermediate system ( ˜ F , ˜ M ), if we let ˜ R : M → N to be the first returntime, such that F x = ˜ F ˜ R ( x ) ( x )Then for any n ≥
1, ( ˜ R = n ) = ∪ ∞ m =1 C m,n Now we use Lemma 10, to get(5.6) µ ( ˜ R ≥ n ) = X m ≥ X k ≥ n µ ( C m,k ) ∼ n − − β − This implies that η = 1 + β − = β +2 β − for system ( ˜ F , ˜ M ). Let S F = (˜ S ∩ M ) [ ∪ m ≥ (cid:0) ∂U m ∪ F − ∂U m (cid:1) For a fixed m ≥
1, let W be an unstable curve contained in C m \ S F . It follows from(3.11) that the curvature K is bounded away from zero in cell C m . According to (4.13)15he expansion factor at x ∈ W becomes very large, as cos ϕ gets small. This implies thatthe expansion factors along W maybe highly nonuniform. To overcome this difficulty wedivide M into horizontal strips as introduced in [1, 2]. More precisely, one divides M intocountably many sections (called homogeneity strips ) defined by H k = { ( r, ϕ ) ∈ M : π/ − k − < ϕ < π/ − ( k + 1) − } and H − k = { ( r, ϕ ) ∈ M : − π/ k + 1) − < ϕ < − π/ k − } for all k ≥ k and(6.1) H = { ( r, ϕ ) ∈ M : − π/ k − < ϕ < π/ − k − } , here k ≥ S H be the collection of all boundaries of these homogeneity strips. Then F − S H isa countable union of stable curves that are almost parallel to each other and accumulateon the singular curves in ˜ S . This implies that there are three sequences of singularcurves in each m -cell C m : the sequences { s m,k } and s ′ m,k accumulate to the curve w sm andthe third one belongs to F − S H that converges to s m +1 . Define(6.2) S = S H ∪ S F , S − = F S ∪ S . Clearly M \ S is a dense set in M . Whenever we say ‘the singularity of F ’, we refer to S although curves in S H are really artificial singular curves for the map.Since billiards have singularities, if the orbit of x approaches to the singularity set S too fast under F , then x may not have a stable manifold. For the finite horizon case,one can show that m -a.e. x ∈ M does have a stable (resp. unstable) manifold. Howeverthe situation is much complicated here as we have more accumulated sequences of singularcurves. Indeed we will first show that small neighborhood of the singular set has smallmeasure. Lemma 11.
For any δ > , the δ − neighborhood of S ± has measure: (6.3) µ ( B δ ( S ± )) ≤ Cδ β β − Here B δ ( S ± ) = { x ∈ M : d M ( x, S ± ) ≤ δ } for any δ > , and C > is a constant.Proof. We first need to find the smallest m δ such that ∪ m ≥ m δ C m ⊂ µ ( B δ ( S )). Accordingto Lemma 5, the height of C m is approximately O ( m − ). Thus we get m δ = δ − / . Thisimplies that µ ( ∪ m ≥ m δ C m ) = O ( m − δ ) = O ( δ )16ext we consider for m < m δ . In each C m , there are two sequences of cells bounded bysingular curves in S : one of which consists of curves γ k ⊂ F − { H k } that approaching s m +1 ;and the other is { s m,k } that approaching w nm . Case I.
We assume W to be a vertical curve intersecting a converging sequence γ k ⊂ F − { H k } that approaching s m +1 . Then F W is contained in the smaller region C ′′ m in C m . Let W k be a vertical curve completely stretch in F − ( H k ). Then the expansion on W k is approximately τ / cos ϕ ∼ mk . Since the ϕ -dimension of F W k is approximately k − . By Lemma 8, we know that the slope of tangent vector of F W k is approximatelycos ϕ β − β − ∼ k − β − β − . Thus | W k | ∼ k − β +1 β − m − Let k m be the smallest integer such that F − H k ∩ C m is not empty. As it follows fromLemma 5 that the height of C m is approximately m − , thus X k ≥ k m | W k | = X k ≥ k m k − β +1 β − m − ∼ m − This implies that k m ∼ m β − β Define k ′ m such that [ k ≥ k ′ m ( F − H k ∩ C m ) ⊂ µ ( B δ ( S ± ))Then k ′ − β +1 β − m m − = δ , which implies that k ′ m = (cid:18) mδ (cid:19) β − β − Since H k is contained in C ′′ m , so it has ϕ -dimension ∼ k − , length ∼ | F W k | ∼ k − − β − anddensity ∼ k − . Then X k ≥ k ′ m µ ( F − H k ∩ C m ) ∼ X k ≥ k ′ m µ ( H k ∩ C ′′ m ) ∼ ck − − β − m ′ This implies that m δ X m =1 X k ≥ k ′ m µ ( F − H k ∩ C m ) + X m ≥ m δ µ ( C m ) ≤ Cδ ase II. We assume a vertical curve W intersects a sequence { s m,k } that approaching w nm . Assume δ = h m,k ′′ m = m − β − β − k ′′− β − β − m Then k ′′ m = (cid:16) δ − m − β − β − (cid:17) β − β − = 1 δ β − β − m β − β − By Lemma 10, the measure of these sets in C m satisfies: X k ≥ k ′′ m µ ( C m,k ) ≤ Cm − − β − k β − β m ′′ = Cm − − β − δ β β − Combining the above facts, we have µ ( B δ ( S ± )) ≤ m δ X m =1 X k ≥ k ′ m µ ( F − H k ∩ C m ) + X k ≥ k ′′ m µ ( C m,k ) + X m ≥ m δ µ ( C m ) ≤ Cδ β β − Since we have added the boundaries of the homogeneity strips and their preimages inthe singular set, the above results implies that for any δ >
0, the δ − neighborhood of S ± has measure:(6.4) µ ( B δ ( S ± )) = O ( δ a )with a = β β − . For any x ∈ M , let r σ ( x ) = d W σ ( x ) ( x, ∂W σ ( x )), where W σ ( x ) is the stable(resp. unstable) manifold that contains x , for σ ∈ { s, u } . Lemma 12.
For any small δ > , the set ( r s ( x ) < δ ) has measure: (6.5) µ ( r s ( x ) < δ ) ≤ Cδ a Proof.
It follows from hyperbolicity there exists c > δ > r s ( x ) < δ ) ⊂ ∪ n ≥ F − n B cδ Λ − np ( S − ) . We apply the measure µ and get µ ( r s ( x ) < δ ) ≤ X n ≥ F n µ ( B cδ Λ − np ( S − )) ≤ X n ≥ µ ( B cδ Λ − np ( S − )) ≤ Cδ a for some C > M has a regular stable (resp.unstable) manifold and there are plenty of reasonable long stable (resp. unstable) manifoldsfor F . 18 Exponential decay rates for the reduced system
According to the general scheme proposed in Section 2, we first need to check condition ( F1 ),i.e. to prove that the reduced system ( F, M, ˆ µ ) enjoys exponential decay of correlations.Here we use a simplified method to prove exponential decay of correlations for our reducedbilliard map. It is mainly based on recent results in [22, 4, 10]. Lemma 13. ( One-step expansion estimate ) Assume β ∈ (2 , ∞ ) , let W be a shortunstable curve in M and { W i } be the collection of smooth components in W . Then (7.1) lim inf δ → sup W : | W | <δ X i ≥ | W i || F W i | < , where the supremum is taken over unstable curves W ⊂ M \ S and Λ i , i ≥ , denote theminimal local expansion factors of the connected component W i under the map F .Proof. Let W ⊂ M be an unstable curve. Since the upper bound of (7.1) only achieveswhen W intersects one of the accumulating sequences. Case I.
First we assume W crosses some s m +1 , the boundary of cell C m , for some m ≥ F − { ∂ H k } . Define W k = W ∩ F − H k . Let x ∈ W k , and x = F x , then the expansion factor | F W k || W k | ∼ Λ k ( x ) ∼ τ K ( x )cos ϕ ≥ c mk m − β − ≥ cm β − k On the other hand, since for x = ( r, ϕ ) ∈ M m , by (3.11), the curvature K ( r ) ≤ Cm β − .Thus we also get the upper bound for Λ k ≤ Cm β k . Since by Lemma 8, the slope of F W k is approximately K ( x ), thus | F W k | ∼ k − / K ( x ) ∼ k − β +1 β − , where we have used (4.7), as K ( x ) ∼ cos ϕ β − β − ∼ k − β − β − Thus | W k | ≥ | F W k | / Λ k ≥ m − β k − − β +1 β − . This enable us to find k m – the smallest k suchthat F − H k intersects W : | W | = X k ≥ k m | W k | ≥ m − β k − ββ − m which implies that 1 k m ≤ (cid:16) m β | W | (cid:17) β − β X k ≥ k m k ≤ Cm − β k − m ≤ C | W | β − β m − β − + β − β ≤ C | W | β − β Thus by taking | W | small, we can make the above sum < Case II.
Next we consider the case when W only intersects { s m,k } . By (5.2), we can find k ′′ m – the smallest k such that W intersects s m,k : | W | ∼ X k ≥ k ′′ m h m,k m − β − ∼ m − β +1 β − k ′′− ββ − m Then k ′′ m = (cid:16) | W | − m − β +1 β − (cid:17) β − β = 1 | W | β − β m ( β +1)( β − β − β Note that W ⊂ C m , which also implies that | W | < m − , i.e. m < | W | − . By Proposition9 the expansion factor satisfies(7.2) Λ k := | F W k | / | W k | ≥ cm β − k β − β − Combining the above facts, we have X k ≥ k ′′ m Λ − k ≤ C X k ≥ k ′′ m m β − k β − β − ≤ C m β − | W | < C p | W | Thus by taking | W | small, we can make the above sum < Case III.
Finally we consider the case when W intersects countably many C m . Let m be the smallest m such that W ∩ C m is empty. Then | W | ∼ m − β . For any m > m , let W m,k = C m ∩ W ∩ F − H k , with k ≥ k m = m β − β and Λ m,k be the expansion factor on W m,k .Then by (7.2). X m ≥ m X k ≥ k m Λ − m,k ≤ X m ≥ m X k ≥ k m m β − k β − β − ≤ C | W | Again by taking | W | small, we can make the above sum < W , a point x ∈ W and an integer n ≥
0, we denote by r n ( x ) thedistance between F n x and the boundary of the homogeneous component of F n W containing F n x . Clearly r n ( x ) is a function on W that characterize the size of smooth components20f F n W . We first state the Growth lemma, proved in [13], which is key in the analysishyperbolic systems with singularities. It expresses the fact that the expansion of unstablecurves dominates the cutting by singular curves, in a uniform fashion for all sequences.The reason behind this fact is that unstable curves expand at a uniform exponential rate,whereas the cuts accumulate at only finite number of singular points. The following GrowthLemma can be derived directly from Lemma 13 – the one-step expansion estimates, see [8],[13] for details. Lemma 14. (Growth lemma). There exist uniform constants C g , c > and ϑ ∈ (0 , suchthat, for any unstable curves W and n ≥ : m W ( r n ( x ) < ε ) ≤ C g ( ϑ n m W ( r ( x ) < ε ) + c | W | ) ε This lemma implies that for n large enough, such as n ≥ n W := | log | W | / log ϑ | , one has(7.3) m W ( r n ( x ) < ε ) ≤ C g | W | ε In other words, after a sufficiently long time n ≥ n W , the majority of points in W have theirimages in homogeneous components of F n W that are longer than 1 / (2 C g ), and the familyof points belonging to shorter ones has a linearly decreasing tail.In [4, 10, 13], the following lemma was proved. Lemma 15.
If the induced billiard map F satisfies (7.1). Then there is a horseshoe ∆ ⊂ M such that (7.4) µ (cid:0) x ∈ M : R ( x ; F, ∆ ) > m (cid:1) ≤ Cθ m ∀ m ∈ N , for some θ < , where R ( x ; F, ∆ ) is the return time of x to ∆ under the map F . Thusthe map F : M → M enjoys exponential decay of correlations. Hence we conclude that for β ∈ (2 , ∞ ), the return map F : M → M has exponentialmixing rates by above Lemma, thus condition ( F1 ) is verified. Moreover, Lemma 4 followswith the decay rates given by (2.2). Thus for both systems ( F , M ) and ( ˜ F , ˜ M ), we knowthe decay rates is of order O ( n − η ), using Lemma 4. Next we will improve the decay rates. A general strategy for estimating the correlation function C m ( f, g, F , µ ) for systems withweak hyperbolicity was developed in [10, 12, 7].We first prove two results that will be need in the proof of the main Theorems.21 emma 16. Let D n ( a ) = { x ∈ M : R ( x ) ≥ n − a } , with a ∈ [0 , β ] . Then for any n ≥ , µ ( D n ( a ) | F ( R = n )) ∼ n βa − β − In addition the conditional expectation E ( R ( F x ) | x ∈ M n ) ∼ n β − β . It follows from (5.3) that ( R = n ) is essentially dominated by C n, = ( R = n ) ∩ M n .Thus the proof of the above lemma directly follows from Lemma 12 in [23], which we willnot repeat here. Proposition 17.
There exist e = β , such that for any large m , there exists E m ⊂ ( R = m ) ,with µ (( R = m ) \ E m ) ≤ m − e µ ( R = m ) , and any x ∈ E m , F x, F x, ..., F b ln m x all belong tocells with index less than m − e .Proof. It follows from Lemma 16,(8.1) ∞ X n = m − β µ (( R = n ) | F ( R = m )) = O ( m − β − )Since C m, dominates C m and ( R = m ), we only need to deal with C m, and its iterations.Note that C m, has dimension ≍ m − in the unstable direction, dimension ≍ m − β in thestable direction, and measure µ ( C m, ) ≍ m − . We first foliate C m, with unstable curves W α ⊂ C m, (where α runs through an index set A ). These curves have length | W α | ≍ m − .Let ν m := µ ( C m, ) µ | C m, be the conditional measure of µ restricted on C m, . Let W = ∪ α ∈ A W α be the collection of all unstable curves, which foliate the cell C m, . Then we can disintegratethe measure ν along the leaves W α . More precisely, for any measurable set A ⊂ C m, , ν m ( A ) = Z A ν α ( W α ∩ A ) dλ ( α )where λ is the probability factor measure on A . For each unstable curve W α ∈ W , if F l W α crosses D m := D m ((2 β ) − ), then F l W α is cut into pieces by the boundary of cells in D m .Next we use Lemma 14, notice that if F l ( x ) ∈ C n, , for l = 1 , · · · , b ln m , n > m − β , thenthe length of the largest unstable manifold is ∼ m − − β ) = m − β − β . So applying (7.3)with δ = m − β − β . According to the growth lemma 14, there exists θ ∈ (0 , F l ∗ ν m ( D m ) ≤ cθ l F ∗ ν m ( D m ) + C z m − β − β . By (8.1), we know that F ∗ ν m ( D m ) ≤ Cm − β − ε = β . Thus by (8.2), we get F l ∗ ν m ( D m ) ≤ cθ l m − β − + C z m − β − β Now we sum for l = 1 , · · · , b ln m to get b ln m X l =1 F l ∗ ν m ( D m ) ≤ cθ l m − β − + C z m − β − β ≤ Cm − β for b large. This also implies that for any large m , there exists E m ⊂ ( R = m ), with µ (( R = m ) \ E m ) ≤ m − β µ ( R = m )and any x ∈ E m , F x, F x, ..., F b ln m x all belong to cells with index less than m − β .Now we are ready to prove Theorem 1.The tower in M can be easily and naturally extended to M , thus we get a the Young’stower with the same base ∆ ⊂ M ; and a.e. point x ∈ M again properly returns to ∆ under F infinitely many times. Consider the return times to M under F for x ∈ M . According toLemma 10,(8.3) µ ( x ∈ M : R ( x ) > n ) ∼ n ∀ n ≥ m ≥ x ∈ M denote r ( x ; m, M ) = { ≤ i ≤ m : F i ( x ) ∈ M } Let A m = { x ∈ M : R ( x ; F , ∆ ) > m } ,B m,b = { x ∈ M : r ( x ; m, M ) > b ln m } , where b > µ ( A m ∩ B m,b ) ≤ C · m θ b ln m . Choosing b = − β/ ln θ , then(8.4) const · m θ b ln m ≤ const · m θ − θ ln m = const · m − β . 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’ ∆ of Young’s tower. Ourgoal is to show that µ ( A m \ B m,b ) = O ( m − ).Let I = [ n , n ] be the longest interval, within [1 , m ], between successive returns to M .Without loss of generality, we assume that m − n ≥ n , i.e. the leftover interval to theright of I is at least as long as the one to the left of it (because the time reversibility ofthe billiard dynamics allows us to turn the time backwards). Due to Lemma 17, for largemeasure of typical points y ∈ M | I | we have F t ( y ) ∈ M m t where m t decreases exponentiallyfast. So there exists c >
0, such that m/ ≤ | I | + b | I | − e ln | I | ≤ c | I | which gives | I | ≥ m c .Let G m = { x ∈ A m \ B m,b : | I | ≥ m c } . Thus it is enough to estimate the size of G m .Since for any x ∈ G m , one of its forward images belongs to m | I | with | I | ≥ m c . Applying thebound (5.4) to the interval I gives(8.5) µ ( G m ) ≤ C m · m · m − = Cm − (the extra factors of m must be included because the interval I may appear anywhere withinthe longer interval [1 , m ], and the measure µ is invariant).In terms of Young’s tower ∆, we obtain(8.6) µ ( x ∈ ∆ : R ( x ; F , ∆ ) > m ) ≤ Cm − , ∀ m ≥ F , ˜ M ). Note that by (5.6), we know that measure oflevel set of the return time function for ˜ F satisfies:(8.7) µ ( ˜ R ≥ n ) = X m ≥ X k ≥ n µ ( C m,k ) ∼ n − − β − Since Section 8 have verified ( F1 ) for the system ( ˜ F , ˜ M ), it is enough to verify Proposition17 for ( ˜ F , ˜ M ) to get the improved bound. Since the estimation is very similar, althoughwith different value of e , we will not repeat here.For Theorem 3. It directly follows from results in [7], once we verify conditions ( F1 )-( F2 )and Proposition 17, which is equivalent to condition (H2)(b) in [7].24 Proof of Proposition 9
The proof is similar to the proof of Proposition 1 in [11]. For completeness, we provide adetailed proof using similar notations.Assume W ⊂ C m,k is an unstable curve. Then the expansion factor (in the Euclidianmetric) along x ∈ W satisfies(9.1) Λ( x ) = k Y j =0 (cid:0) τ ( x j ) B ( x j ) (cid:1) cos ϕ j cos ϕ j +1 ≥ Cτ ( x k ) K ( x k )cos ϕ k k =1 Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) where x j = ( r j , ϕ j ) = F jm ( x ), for j = 0 , · · · , k −
1, with x = x . Note that τ ( x j ) ∼ m, cos ϕ j ∼ m − for j = 1 , · · · , k . Denote Λ m,k ( x ) = Q k − j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) as the expansionfactor in the p -metric along all remaining collisions in the series. Thus(9.2) Λ( x ) ≥ Cm K m Λ m,k ( x ) ≥ Cm · m − ββ − Λ m,k ( x ) ≥ Cm β − Λ m,k ( x )Next we will estimate the lower bound for Λ m,k ( x ).Note that B ( x j ) satisfies the recurrent formula(9.3) B ( x j ) = 2 K ( r j )cos ϕ j + 1 τ ( x j − ) + 1 / B ( x j − ) , Since τ ( x j − ) is the distance between x j − and x j in the unfolding table ˜ Q . Clearly, τ ( x j ) ∼ m , for any j = 1 , ..., k − K ( r j ) ∼ β ( β − | r j | β − . Now we see that in order to estimate the expansion factor Λ( x ) given by (9.2), it is enoughto estimate the r -coordinates { r j } along the trajectory of x ∈ C m,k .Let x be a point in C m,k (the case x ∈ C ′ m,k is easier and will be treated later). Denote s x as the distance from x to γ y m in Q . Define a variation flow { ψ ( s, t ) : s ∈ [0 , s x ] , t ∈ ( −∞ , ∞ ) } of γ y m such that ψ (0 , t ) has the same trace as γ y m 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 J ( t ) is also called a (generalized) Jacobi field along γ x . Denote by J j the correspondingJacobi vector based at x i . Notice ˙ J = dJdt is also a vector field along γ x . Correspondingly,we denote by v j = − ˙ J j , which also represents the angle made by the orbit of F jm ( y m ) andthat of x j = F jm ( x ).Note that ( βr β − j ,
1) is the inward normal vector to ∂Q at the point r j . Geometric con-siderations (more precisely, the generalized Jacobi equations) yield the following relations: v j − v j +1 = 2 arctan( βr β − j +1 ) J j − J j +1 = m tan v j + 2( r βj + r βj +1 ) tan v j . (9.5) 25lso notice r j +1 = J j +1 cos ϕ j +1 . Since C ′ m,k contains points whose images will be trapped in the window during the next k − F m . Thus using Taylor expansion we obtain v j − v j +1 = 2 βr β − j +1 − R v,j +1 r j − r j +1 = m v j + R r,j (9.6)where(9.7) R v,j +1 ∼ r β − j +1 > R r,j ∼ mr βj v j + mv j > R v,j +1 and R r,j is guaranteed by the smallness of ε and by geometry, v j ≪ r β − j ).Note that formula (9.6) differs from formula (4.8) in [11] only by a constant m /
2. Asa result, the proofs in the Appendix of that paper can be directly applied to our modelwithout much changes. In fact we can get an idea to understand the above lemma fromthe following estimations. For j large enough, the solutions of the above systems can beapproximated by the solutions of the following differential equation:(9.9) dv/dt = 2 βr β − ( t ) and dr/dt = m v ( t ) . Let k ′ be uniquely defined by r k ′ +1 ≥ r k ′ . First we consider the interval 1 ≤ j ≤ k ′ , i.e.where { r j } is decreasing. Note that both { r j } and { v j } are decreasing sequences of positivenumbers for j = 1 , . . . , k ′ . Using the equation (9.9), one can show that for any j = 1 , . . . , k ′ the following relation is always true:(9.10) m v j ∼ r βj and r β − j ∼ (cid:16) ( β − jm + r − β/ (cid:17) − ∼ (cid:16) ( β − jm + m β − β − (cid:17) − , where | r | > m − β − is the initial position for this sequence. Lemma 18.
Let k ′′ ∈ [1 , k ′ ] be uniquely defined by the condition (9.11) v k ′′ − > v k ′ > v k ′′ . hen for all k ′ < j ≤ k ′′ we have (9.12) r j ∼ m ( k ′ − j ) v k ′ And (9.13) k ′ − k ′′ ∼ m β − β v − β k ′ ∼ m β r − β k ′ Proof.
Due to (9.6), for any j ∈ [ k ′′ , k ′ ) we have m v k ′ ≤ r j − r j +1 ≤ m v j ≤ m v k ′′ ≤ m v k ′ This implies(9.14) m ( k ′ − j ) v k ′ ≤ r j ≤ m ( k ′ − j + 1) v k ′ . Now we get (9.12) and r k ′ ∼ m v k ′ . (9.6) implies that m β − ( k ′ − j ) β − v β − k ′ ≤ v j − v j +1 ≤ βr β − j ≤ β β − m β − ( k ′ − j ) β − v β − k ′ therefore v j ∼ v k ′ + m β − ( k ′ − j ) β v β − k ′ Substituting j = k ′′ , then j = k ′′ − m β − ( k ′ − j ) β v β − k ′ ∼ v k ′ Taking j = k ′′ in (9.14), we get (9.13)We note that the above lemma implies(9.15) r βk ′′ ∼ m v k ′ , r k ′ ∼ m v k ′ hence r k ′′ ≪ ε m and thus k ′′ ≫
1. Next we consider the case 1 < j ≤ k ′′ . Lemma 19.
For all < j ≤ k ′′ we have (9.16) r j ∼ ( mj ) − β . and (9.17) mk ′′ ∼ ( mv k ′ ) − ββ . Furthermore, (9.18) k ′′ ∼ k ′ ∼ k and r k ′ ∼ m − β k ββ − roof. Denote z j = r β − j . Then (9.6) and the mean value theorem imply z j − z j +1 ∼ m β − J β − j ( J j − J j +1 ) ∼ m β − J β − j (2 v j ) ∼ z j (we used the relation in (9.10)). Now let Z j = 1 /z j , then Z j +1 − Z j ∼ Z j +1 /Z j Note that by (9.6), r j − r j +1 ∼ r β/ j ≪ r j Hence Z j +1 − Z j ∼
1. Since r ≥ ε m ,(9.19) Z ≤ ε − β − m = const. m β − β − , and we obtain(9.20) Z j ∼ mj and z j ∼ ( mj ) − , which proves (9.16). Now (9.17) is immediate due to (9.15). Equation (9.17) also imply k ′′ ∼ k ′ ∼ k and mk ′ ∼ ( mv k ′ ) − ββ . Lemma 20.
For all ≤ j ≤ k ′′ we have (9.21) r β − j ≥ D (cid:18) j + C ln j + C j (cid:16) jk (cid:17) ββ − + C (cid:19) − where D = m ( β − and C , C , C > are some constants. By (9.4) we have proved the following estimation on K ( r j )cos ϕ j . Corollary 21.
For all ≤ j ≤ k ′′ we have (9.22) 2 τ ( r j , ϕ j ) K ( r j )cos ϕ j ≥ D (cid:20) j + C ′ ln j + C ′ j (cid:16) jk (cid:17) ββ − + C ′ (cid:21) − where D = β ( β − β − . Note that τ ( r j , ϕ j ) K ( r j ) / cos ϕ j ∼ j − , which does not depend on m . Now we use therelation (9.3) and get the estimation for τ ( x j ) B ( x j ).28 emma 22. For all ≤ j < k ′′ we have (9.23) τ ( x j ) B ( x j ) ≥ Aj + C ′ ln jj , where A > satisfies A − A = β ( β − β − , hence A = β − β − . Now we are ready to estimate the expansion factor Λ m,k ( x ) given by (9.2). Lemma 23. (9.24) k ′′ Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) ≥ Ck β − β − where C > is a constant.Proof. Note that τ ( x j ) > m . Hence, due (9.23), we haveln " k ′′ − Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) > k ′′ X j =0 ( Aj + C ′ ln jj )with some large constant C ′ >
0. Therefore,ln " k ′′ Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) > A ln k ′′ + const > A ln k + const , where the last inequality follows from (9.18). Lastly, note that A = β − β − , which completesthe proof of the lemma.Next we are ready to estimate the expansion factor Λ m,k ( x ). Lemma 24. (9.25) Λ m,k ( x ) = k − Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) ≥ Ck β − β − − where C > is a constant.Proof. We will use the time-reversibility of the billiard dynamics. Let V u and V s be twounit vectors (in the p-norm) tangent to the unstable and stable manifolds, respectively, atthe point x . Since B ( x ) − = O ( m − ), (4.1) implies that the slope of the vector V uk ′ is dϕdr = cos ϕ B ( x ) − + K ( r ) ≤ c m . V u makes an angle of order cm − with the horizontal r -axis. By the timereversibility, the vector V s makes an angle less than − cm − with the horizontal r -axis. Thusthe area of the parallelogram Π spanned by V u and V s is of order O ( m − ).Consider the parallelogram Π k ′ = D x F k ′ m (Π) spanned by the vectors V uk ′ = D x F k ′ m ( V u )and V sk ′ = D x F k ′ m ( V s ). Since the map F k ′ m preserves the measure dµ = cos ϕ dr dϕ , wehave cos ϕ k ′ Area(Π k ′ ) = cos ϕ Area(Π) . Note that cos ϕ ≤ cm − and cos ϕ k ′ ≈ m − , henceArea(Π k ′ ) ≤ Area(Π) cos ϕ cos ϕ ′ k ≤ cm − . On the other hand, Area(Π k ′ ) = | V uk ′ | p | V sk ′ | p sin γ k ′ where | V uk ′ | p and | V sk ′ | p denote the lengths of these vectors in the p-norm and γ k ′ denotes theangle between them.Next we estimate γ k ′ . It easily follows from (9.23) that B ( x k ′ ) ∼ (cid:18) km + 1 B ( x ) (cid:19) − ∼ km Now by (9.13) and (9.18) we have(9.26) cos ϕ k ′ B − ( x k ′ ) ≥ c m k , K ( r k ′ ) ≥ c m k β So (9.3) implies that dϕdr ≥ Cm k for some constant C >
0. Hence sin γ k ′ ≥ Cm k for some constant c >
0, and we obtain | V uk ′ | p | V sk ′ | p ≤ c m km = c k for some constant c >
0. Note that | V uk ′ | p = Λ (1) m,k ( x ) | V u | p ∼ Λ (1) m,k ( x ) . where Λ (1) m,k ( x ) is the expansion factor corresponding to collisions from 1 to k ′ , withΛ (1) m,k ( x ) := k ′ Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1)
30y the time reversibility of the billiard dynamics, the contraction of stable vectors duringthe time interval (0 , k ′ ) is the same as the expansion of the corresponding unstable vectorsduring the time interval ( k ′ , k − | V sk ′ | p ∼ Λ (2) m,k ( x ) − | V s | p ∼ Λ (2) m,k ( x ) − , where Λ (2) m,k ( x ) is the expansion factor corresponding to collisions from k ′ + 1 to k −
1, withΛ (2) m,k ( x ) := k − Y j = k ′ +1 (cid:0) τ ( x j ) B ( x j ) (cid:1) Therefore,(9.27) Λ (2) m,k ( x ) > Λ (1) m,k ( x ) ck for some constant c >
0. Now we combine with (9.24) and getΛ m,k ( x ) ≥ Λ (1) m,k ′ ( x )Λ (2) m,k ′ ( x ) ≥ Ck β − β − − . This proves Lemma 9.25 for W ⊂ C m,k .We finally consider the remaining case W ⊂ C ′ m,k . In that case k ′ can be defined as theturning point, i.e. by r k ′ < r k ′ − and r k ′ < r k ′ +1 . Observe that if x ′ = ( r ′ , ϕ ′ ) ∈ C ′ m,k , thenthere exists another point x = ( r, ϕ ) ∈ C m,k with r = r ′ and ϕ < ϕ ′ , whose trajectory goesthrough the window. Since ϕ ′ < ϕ , it follows that the r -coordinate of the point F jm ( x ) willbe always smaller than the r -coordinate of the point F jm ( x ′ ), for all 1 ≤ j ≤ k −
2. Thisobservation and the bound (9.21) that we have proved for J j implies that the same boundholds for r k and for all 1 ≤ j ≤ k ′′ . The rest of the proof of Lemma 9.25 for x ′ ∈ C ′ m,k isidentical to that of the case x ∈ C m,k .Now we prove Proposition 9 using (9.2). Combining with the above lemma togetherwith (9.2), we have shown(9.28) Λ( x ) = k − Y j =0 (cid:0) τ ( x j ) B ( x j ) (cid:1) cos ϕ j cos ϕ j +1 ≥ Cm β − k β − β − − where C > W is a vertical curve, since by the differential (3.1), dr /dϕ = τ / cos ϕ , itsexpansion factor for the collision under F m does not depend on K . This implies that theexpansion factor along x ∈ W satisfies(9.29) Λ( x ) = k − Y j =0 (cid:0) τ ( x j ) B ( x j ) (cid:1) cos ϕ j cos ϕ j +1 ≥ Cτ ( x )cos ϕ k k − Y j =1 (cid:0) τ ( x j ) B ( x j ) (cid:1) ≥ Cm k β − β − − Acknowledgement . This paper is written in memory of Professor Nikolai Chernov. Theauthor is also partially supported by NSF CAREER Grant (DMS-1151762) and the Si-mons Fellowship. The author also would like to thank Dmitry Dolgopyat for many helpfuldiscussions on the maps considered in this paper.
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