Billiards with polynomial mixing rates

By Nikolai I. Chernov, Hong-Kun Zhang. 2004

143 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ddcdbb88e4fe0de885458aeeaf736e8af533bd7f

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Dispersing Billiards with Cusps: Slow Decay of Correlations (2007, methodology): “Lastly one uses the following theorem proven in [ CZ05a , Section 3]:”
Martingale–coboundary decomposition for families of dynamical systems (2016, background): “By [19], the quotient map T (obtained by quotienting out stable manifolds) is nonuniformly expanding with inducing time τ ∈ Lp for all p < 3.”
Large deviations for nonuniformly hyperbolic systems (2008, background): “Other examples with β = 1 include Bunimovich-type stadia and certain classes of semidispersing billiards; see [3, 6, 22].”