Decay of correlations for non-uniformly expanding systems with general return times. (English) Zbl 1332.37025

Melbourne, Ian; Terhesiu, Dalia

Ergodic Theory Dyn. Syst. 34, No. 3, 893-918 (2014).

Let \(T\) be a measurable transformation on \(X\) and \(\phi:Y\to \mathbb{N}\) be a measurable map on a subset \(Y\subset X\) with \(Fy:=T^{\phi(y)}y\in Y\). Note that \(\phi\) may not be the first return time map of \(T\) on \(Y\). This induces a measurable map \(F\) on \(Y\). Let \(\mu\) be the \(F\)-invariant and ergodic probability measure, and \(\mu_X\) be the lifted measure of \(\mu\) on \(X\), which is \(T\)-invariant and ergodic. In this paper the authors give a unified treatment on how to derive the decay of correlations of the system \((T,\mu_X)\) from the properties of \((F,\mu)\) and the tail information of \(\mu(\phi>n)\).
Several interesting applications are given. In the case of a Young tower, that is, \(F\) is uniformly expanding with nice distortion properties, they prove that the decay rate on (piecewise) Holder functions is completely determined by the tail information of \(\phi\) and hence is uniform. More generally, they consider the systems with “excellent” inducing schemes and with “good” inducing schemes. Under some natural assumptions, they prove that the decay rates for these systems are completely determined by the tail information and hence are uniform.

Reviewer: Pengfei Zhang (Oxford, MS)

MSC:

37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37A25	Ergodicity, mixing, rates of mixing
37D50	Hyperbolic systems with singularities (billiards, etc.) (MSC2010)

Keywords:

correlation; mixing; decay rate; uniform decay rate; Young tower; nonuniform hyperbolicity; return time; inducing scheme; exchangeable observables

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