Statistical properties of mostly contracting fast-slow partially hyperbolic systems

TL;DR

This paper analyzes the statistical behavior of a class of partially hyperbolic systems on the torus, classifying SRB measures and establishing exponential decay of correlations with explicit bounds.

Contribution

It provides a precise classification of SRB measures and their statistical properties for a family of partially hyperbolic systems with small perturbations.

Findings

Classification of SRB measures for the system.

Proof of exponential decay of correlations.

Explicit bounds on decay rates.

Abstract

We consider a class of $\mathcal C^{4}$ partially hyperbolic systems on $\mathbb T^2$ described by maps $F_\varepsilon(x,\theta)=(f(x,\theta),\theta+\varepsilon\omega(x,\theta))$ where $f(\cdot,\theta)$ are expanding maps of the circle. For sufficiently small $\varepsilon$ and $\omega$ generic in an open set, we precisely classify the SRB measures for $F_\varepsilon$ and their statistical properties, including exponential decay of correlation for H\"older observables with explicit and nearly optimal bounds on the decay rate.