Exponential mixing for singular skew-products

Abstract

We study skew-products of the form $(x,u)\mapsto (fx,u+\phi (x))$ where $f$ is a non-uniformly expanding map on a manifold $X$ and $\phi :X\to {\mathbb{S}}^1$ is piecewise ${\mathcal{C}}^1$. If the systems satisfies mild assumptions (in particular singular behaviour of $\phi$ 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.

References in corpus (3)

• Polynomial decay of correlations for flows, including Lorentz gas examples
• Statistical properties for flows with unbounded roof function, including the Lorenz attractor
• Sharp Statistical Properties for a Family of Multidimensional NonMarkovian Nonconformal Intermittent Maps