Differentiable Conjugacy of Anosov Diffeomorphisms on Three Dimensional Torus

TL;DR

This paper establishes a criterion for when two C^2 Anosov diffeomorphisms on a three-dimensional torus are conjugate, based on the eigenvalues of return maps at periodic points, contributing to the understanding of their structural stability.

Contribution

It provides a necessary and sufficient condition for C^1+ conjugacy of Anosov diffeomorphisms near a linear automorphism on the 3-torus, focusing on eigenvalues of return maps.

Findings

Conjugacy is characterized by eigenvalues of return maps.

C^1+ conjugacy holds if and only if eigenvalues match.

Results apply to diffeomorphisms near a linear hyperbolic automorphism.

Abstract

We consider two C^2 Anosov diffeomorphisms in a C^1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are C^1+ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.