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

Abstract

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