Correlation decay and recurrence estimates for some robust nonuniformly hyperbolic maps

TL;DR

This paper investigates decay of correlations, hitting time distributions, and return time fluctuations for a class of multidimensional non-uniformly hyperbolic maps, establishing exponential decay and distribution properties.

Contribution

It proves quasi-compactness of the Ruelle-Perron-Frobenius operator for a broad class of potentials, leading to new results on decay rates and fluctuations for these maps.

Findings

Exponential decay of correlations for the studied maps.

Exponential asymptotic distribution of hitting times.

Log-normal fluctuations of return times.

Abstract

We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state for a large class of non- uniformly expanding transformations and Holder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space $V_\theta$ of functions with essential bounded variation that strictly contain Holder continuous observables. We deduce that the equilibrium states have exponential decay of correlations. Furthermore, we prove exponential asymptotic distribu- tion of hitting times and log-normal fluctuations of the return times around the average given by the metric entropy.