Limit Theorems in the Stadium Billiard

Bálint, Péter; Gouëzel, Sébastien

doi:10.1007/s00220-005-1511-6

Cited by 75 publications

(137 citation statements)

References 18 publications

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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]. 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).…”

Section: F(x)mentioning

confidence: 99%

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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Section: F(x)mentioning

confidence: 99%

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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“…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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“…This value is reminiscent of the limit variance seen in central limit theorems for the stadium billiard of deterministic billiard dynamics (see [4]). A simple modification of the semicircles contour is shown in Figure 2.3.…”

Section: Examples Of Diffusivity For Geometric Microstructuresmentioning

confidence: 75%

Diffusivity in multiple scattering systems

Chumley¹,

Feres²,

Zhang³

2015

Trans. Amer. Math. Soc.

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We consider random flights of point particles inside n-dimensional channels of the form R k ×B n−k , where B n−k is a ball of radius r in dimension n−k. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator P that is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the "microscopic" scale. Our central concern is the relationship between the scattering properties encoded in P and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. Markov operators obtained in this way are natural (definition below), which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of P . We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of P and compute, in the case of 2-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).

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Limit Theorems in the Stadium Billiard

Cited by 75 publications

References 18 publications

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

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

Large deviations for nonuniformly hyperbolic systems

Diffusivity in multiple scattering systems

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