Decay of correlations for unbounded observables

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 mu. We prove that, under some general assumptions, the correlations decay exponentially as n -> infinity for each pair of piecewise Holder observables f, g is an element of L ( p )(mu) 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).

Automated summary

In this article, the decay rates of correlation functions for a hyperbolic system with singularities that preserves a unique mixing SRB measure are studied. It is proven that, under some general assumptions, correlations decay exponentially as n -> infinity for each pair of piecewise Holder observables and for each p > 1. The exponential decay of autocorrelations of first return time functions for induced maps of various billiard systems is also demonstrated, which is essential in studying the statistical properties of nonuniformly hyperbolic systems with singularities.