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

Abstract

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

The proof of the third item of Proposition 3.2 of original version has a gap