Anosov Endomorphisms on the 2-torus: Regularity of foliations and rigidity

TL;DR

This paper establishes conditions under which Anosov endomorphisms on the 2-torus are smoothly conjugate, linking foliation regularity to conjugacy and characterizing such relationships for specific classes including constant Jacobian cases.

Contribution

It provides new sufficient conditions for smooth conjugacy of Anosov endomorphisms on the 2-torus based on foliation regularity and characterizes conjugacy within certain classes.

Findings

Regularity of foliations implies smooth conjugacy.

Characterization of conjugacy for endomorphisms with constant Jacobian.

Conditions for conjugacy between Anosov endomorphisms and their linearizations.

Abstract

We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the 2-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterization of smooth conjugacy between special Anosov endomorphisms and their linearizations.