Decay of correlations for unbounded observables. (18th February 2021)

• Record Type: Journal Article
• Authors: Wang, Fang; Zhang, Hong-Kun; Zhang, Pengfei
• Is Part Of: Nonlinearity. Volume 34:Number 4(2021)
• Page Start: 2402
• Page End: 2429
• Publication Date: 2021-02-18
• Subjects: decay of correlations -- coupling method -- hyperbolic systems with singularities; 37A25, 37D40; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abbab2
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this article, we study the decay rates of the correlation functions for a hyperbolic system T : M → M with singularities that preserves a unique mixing SRB measure μ . We prove that, under some general assumptions, the correlations decay exponentially as n → ∞ for each pair of piecewise Hölder observables f, g ∈ L p ( μ ) and for each p > 1. As an application, we prove that the autocorrelations of the first return time functions decay exponentially for the induced maps of various billiard systems, which include the semi-dispersing billiards on a rectangle, billiards with cusps, and Bunimovich stadia. These estimates of the decay rates of autocorrelations of the first return time functions for the induced maps have an essential importance in the study of the statistical properties of nonuniformly hyperbolic systems (with singularities).