Exponential mixing for singular skew-products

TL;DR

This paper proves exponential mixing for a class of skew-products with non-uniformly expanding base maps and piecewise smooth fiber maps, even allowing singularities, extending previous results in dynamical systems.

Contribution

It establishes exponential mixing for skew-products with singular fiber maps, broadening the class of systems known to exhibit this property.

Findings

Exponential mixing is achieved under mild assumptions.

Singular behavior in the fiber map is permitted.

Results extend previous work to include systems with singularities.

Abstract

We study skew-products of the form $(x,u) \mapsto (fx, u + \varphi(x))$ where $f$ is a non-uniformly expanding map on a manifold $X$ and $\varphi: X \to \mathbb{S}^1$ is piecewise $\mathcal{C}^1$. If the systems satisfies mild assumptions (in particular singular behaviour of $\varphi$ is permitted) then we prove that the map mixes exponentially with respect to the unique SRB measure. This extends previous results by allowing singular behaviour in the fibre map.