Exponential speed of mixing for skew-products with singularities

TL;DR

This paper proves that a specific class of smooth skew-product maps with singularities are topologically mixing and exhibit exponential mixing speed, especially when a parameter exceeds a certain threshold.

Contribution

It establishes exponential mixing rates for a class of singular skew-products, extending understanding of their dynamical complexity.

Findings

The map is topologically mixing.

For c > 1/4, the map is mixing with Lebesgue measure.

The mixing speed is proven to be exponential.

Abstract

Let $f: [0,1]\times [0,1] \setminus {1/2} \to [0,1]\times [0,1]$ be the $C^\infty$ endomorphism given by $$f(x,y)=(2x- [2x], y+ c/|x-1/2|- [y+ c/|x-1/2|]),$$ where $c$ is a positive real number. We prove that $f$ is topologically mixing and if $c>1/4$ then $f$ is mixing with respect to Lebesgue measure. Furthermore we prove that the speed of mixing is exponential.