Statistical properties of mostly expanding fast-slow partially hyperbolic systems

TL;DR

This paper studies a class of fast-slow partially hyperbolic systems on the torus, proving existence and uniqueness of physical measures and exponential decay of correlations under small perturbations.

Contribution

It establishes the existence, uniqueness, and statistical properties of physical measures for mostly expanding fast-slow systems, extending previous work on mostly contracting centers.

Findings

Existence and uniqueness of physical measures for small perturbations.

Exponential decay of correlations with explicit bounds.

Applicable to a broad class of $C^4$ partially hyperbolic systems.

Abstract

We consider a class of fast-slow $C^4$ partially hyperbolic systems on $\mathbb{T}^2$ given by $\epsilon$-perturbations of maps $F(x,\theta)=(f(x,\theta),\theta)$ where $f(\cdot,\theta)$ are $C^{4}$ expanding maps of the circle. For sufficiently small $\epsilon$ and an open set of perturbations we prove existence and uniqueness of a physical measure and exponential decay of correlations for sufficiently smooth observables with explicit almost optimal bounds on the decay rate. Our result complements previous work by De Simoi and Liverani, which studied the case of mostly contracting centre.