Stable ergodicity of skew product endomorphisms on $T^2$

TL;DR

This paper investigates the conditions under which skew product endomorphisms on the 2-torus exhibit stable ergodicity, revealing a dichotomy based on the cohomology class of the perturbation function.

Contribution

It establishes a dichotomy for the stable ergodicity of skew product endomorphisms on the torus depending on the cohomology class of the perturbation function.

Findings

Dichotomy between ergodic and non-ergodic behavior based on cohomology class.

Stable ergodicity persists under certain non-skew product perturbations.

Characterization of ergodic stability for a family of endomorphisms.

Abstract

For a family of skew product endomorphisms on closed surface $f_{\phi}(x,y)=\big(l x\ , \ y+\phi(x)\big)$, where $l\in \mathbb{N}_{\geq 2}$ and $\phi: S^1\to \mathbb{R}$ is a $C^r\ (r>1)$ function, we get a dichotomy on the cohomology class of $\phi$ and the $C^1$-stable ergodicity of $f_{\phi}$, where the perturbation may not be a skew product endomorphism.