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

TL;DR

This paper proves that certain partially hyperbolic endomorphisms on the 2-torus are topologically conjugate to expanding linear endomorphisms if and only if they are area-expanding, with conjugacy regularity depending on smoothness.

Contribution

It establishes a characterization of topological conjugacy for partially hyperbolic endomorphisms on the 2-torus in terms of area-expanding property and smoothness of conjugacy.

Findings

Conjugacy exists if and only if the endomorphism is area-expanding.

The conjugacy is as smooth as the endomorphism's regularity allows, up to real-analytic.

The result applies to endomorphisms homotopic to expanding linear maps with irrational eigenvalues.

Abstract

Assume that $f$ is a $C^r(r\geq 3)$ 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 $C^{\max\{r-3,1\}+\alpha}$. In particular, if $r=\omega$, the conjugacy is $C^{\omega}$.