Exponential Mixing for Skew Products with Discontinuities

Abstract

We consider the skew product $F:(x,u)\mapsto (f(x),u+\tau (x))$, where the base map $f:{\mathbb{T}}^1\to {\mathbb{T}}^1$ is piecewise ${\mathcal{C}}^2$, covering and uniformly expanding, and the fibre map $\tau :{\mathbb{T}}^1\to \mathbb{R}$ is piecewise ${\mathcal{C}}^2$. We show the dichotomy that either this system mixes exponentially or $\tau$ is cohomologous (via a Lipschitz function) to a piecewise constant.

Cited by in corpus (1)

• On the spectra of quenched random perturbations of partially expanding maps on the torus