Billiards with polynomial mixing rates

Abstract

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic -enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.

A Family of Chaotic Billiards With Variable Mixing Rates

“…It is shown in [8] that Theorem 1 follows from (F1) and (F2). Also, the proof of (F1) is reduced in [8] to the verification of the following property of unstable manifolds: Let W ⊂ M denote an unstable manifold (it is a smooth curve since dim M = 2).…”

“…That scheme has been successfully applied in [8] to various classes of chaotic billiards. The scheme is based on finding a subset M ⊂ M where the map F is strongly (uniformly) hyperbolic and the subsequent analysis of the return map F : M → M, which is defined by…”

“…3. Let q 1 = (ε, −g β (ε)) denote one of the four points on ∂Q that border the window |x| < ε, and by q 2 , q 3 , q 4 the other three points ( In order to prove Theorem 1, according to our general scheme [8], we need to establish two properties of the map F described below.…”

Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing

“…The fact that (M, f, µ) can be modeled by a Young tower satisfying (P1)-(P5) can be found in [15,8,4]. Namely, the fact that (P2)-(P3) hold with α = 1 (and so (2)) comes from Propositions 2.1 and 2.3 of [4] and the fact that (1) holds with ζ = 2 is proved in Section 9 of [8]. Finally, because of the continuity and positivity of the density function of µ with respect to the Lebesgue measure, (3) holds for every x ∈ M (See the Appendix for details).…”

Large deviations for nonuniformly hyperbolic systems

“…An interesting problem is to investigate the case β ∈ (0, 1] which occurs in Example 1.3 below for α ∈ [ 1 2 , 1). Other examples with β = 1 include Bunimovich-type stadia and certain classes of semidispersing billiards; see [3,6,22]. Example 1.3 (Intermittency-type maps).…”