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

TL;DR

This work addresses the rigidity of specially partially hyperbolic endomorphisms $f$ on $ abla ext{T}^2$ that are homotopic to an expanding linear map $A$ with irrational eigenvalues. It proves that the natural semi-conjugacy $h$ between $f$ and $A$ is a genuine topological conjugacy precisely when $f$ is area-expanding, and, under the extra assumption that the center bundle $E^c$ is $C^1$, establishes a cascade of smoothness for $h$—from $C^{1+ ext{α}}$ up to $C^{ ext{ω}}$ or $C^ ext{r}$ depending on the differentiability class of $f$. The results extend known rigidity phenomena to the irrational-eigenvalue case by leveraging foliations, bounded-distance properties, Livšic-type cohomology, and Journe-type regularity arguments, yielding both topological and analytic conjugacy conclusions along center and unstable foliations. Collectively, the paper clarifies when derived-from-expanding endomorphisms on $ abla ext{T}^2$ are dynamically equivalent to their linear models and delineates the regularity of the conjugacy under varying smoothness assumptions.

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 $C^{\max\{r-3,1\}+α}$. In particular, if $r=ω$, the conjugacy is $C^ω$.

Figures (3)

• Figure 1: Global product on $\mathbb{R}^2$
• Figure 2: Function $\Phi_{x_0}(x)$
• Figure 3: Cohomology relationship

Theorems & Definitions (53)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.1
• Theorem 1.2
• Remark 1.4
• Theorem 1.3: Main Theorem
• Lemma 2.1
• proof
• Theorem 2.1