Exponential Mixing for Skew Products with Discontinuities

TL;DR

This paper investigates skew product systems with piecewise smooth, expanding base maps and discontinuous fiber functions, establishing conditions under which the system exhibits exponential mixing or reduces to a trivial case.

Contribution

It proves a dichotomy for such skew products: either they mix exponentially or the fiber function is cohomologous to a piecewise constant, extending understanding of mixing in systems with discontinuities.

Findings

System exhibits exponential mixing or fiber function is cohomologous to a piecewise constant.

Provides conditions for exponential mixing in skew products with discontinuities.

Dichotomy clarifies behavior of systems with piecewise smooth, expanding base maps.

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.