Billiards with polynomial decay of correlations

Markarian, Roberto

doi:10.1017/s0143385703000270

Cited by 76 publications

(102 citation statements)

References 9 publications

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“…The same bound on correlations for straight stadia is already obtained by Markarian [19], but we include it here for the sake of completeness.…”

Section: (A)mentioning

confidence: 62%

“…It is based on Young's recent results [24,25] and their extensions by Markarian [19] and one of us [9].…”

Section: Correlation Analysismentioning

confidence: 99%

“…Only in 2004 R. Markarian [19] used Young's method to establish polynomial mixing rates for one class of chaotic billiards -Bunimovich stadia. He showed that the correlations for the corresponding collision map decayed as O n −1 ln 2 n .…”

Section: Introductionmentioning

confidence: 99%

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Billiards with polynomial mixing rates

Chernov

Zhang²

2005

Nonlinearity

119

246

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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.

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“…The same bound on correlations for straight stadia is already obtained by Markarian [19], but we include it here for the sake of completeness.…”

Section: (A)mentioning

confidence: 62%

“…It is based on Young's recent results [24,25] and their extensions by Markarian [19] and one of us [9].…”

Section: Correlation Analysismentioning

confidence: 99%

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Billiards with polynomial mixing rates

Chernov

Zhang²

2005

Nonlinearity

119

246

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“…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].…”

Section: F(x)mentioning

confidence: 91%

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

Pène¹,

Saussol²

2015

Ergod. Th. Dynam. Sys.

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Abstract. We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x, r) converges to a Poisson distribution as the radius r → 0 and after suitable normalization.

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“…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).…”

Section: Introductionmentioning

confidence: 99%

Large deviations for nonuniformly hyperbolic systems

Melbourne

Nicol

2008

Trans. Amer. Math. Soc.

161

245

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Abstract. We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal.In contrast with many treatments of large deviations, our methods do not rely on thermodynamic formalism. Hence, for Hölder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton, as well as polynomial estimates in situations where there is not a unique equilibrium measure.

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Billiards with polynomial decay of correlations

Cited by 76 publications

References 9 publications

Billiards with polynomial mixing rates

Billiards with polynomial mixing rates

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

Large deviations for nonuniformly hyperbolic systems

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