Mixing for some non-uniformly hyperbolic systems

TL;DR

This paper establishes mixing properties and decay of correlations for a broad class of invertible non-uniformly hyperbolic systems, including both infinite and finite measure cases, using operator renewal theory and distribution function spaces.

Contribution

It introduces a novel combination of operator renewal theory and distribution space methods to analyze mixing and correlation decay in invertible non-uniformly hyperbolic systems.

Findings

Proves mixing for invertible systems with infinite measure.

Establishes decay of correlation rates for finite measure non-Markov maps.

Provides sharp mixing rate results in some cases.

Abstract

In this work we obtain mixing (and in some cases sharp mixing rates) for a reasonable large class of invertible systems preserving an infinite measure. The examples considered here are the invertible analogue of both Markov and non Markov unit interval maps. Moreover, we obtain results on the decay of correlation in the finite case of invertible non Markov maps, which, to our knowledge, were not previously addressed. The present method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig [39], with the framework of function spaces of distributions developed in the recent years along the lines of Blank, Keller and Liverani [9].