Derived from expanding endomorphism on ${\mathbb{T}}^2$

Abstract

Assume that $f$ is a $C^r(r\ge 3)$ specially partially hyperbolic endomorphism on the 2-torus which is homotopic to an expanding linear endomorphism $A$ with irrational eigenvalues. We prove that $f$ and $A$ are topologically conjugate, if and only if $f$ is area-expanding. If $f$ is area-expanding and the center bundle is $C^1$, then the topological conjugacy between $f$ and $A$ is $C^{\max \{r-3,1\}+\hat{I}\pm }$. In particular, if $r=\ddot{I}‰$, the conjugacy is $C^{\ddot{I}}‰$.