Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori

Abstract

Let $L$ be a hyperbolic automorphism of ${\mathbb{T}}^d$, $d\ge 3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal{U}$ of $L$. The main result establishes $C^{1+\nu }$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on ${\mathbb{T}}^4$ with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb $C^{1+\nu }$ conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of ${\mathbb{T}}^4$ $C^1$-close to the original example.

arXiv commentTheorem B as stated was wrong in version 1. In this version the statement and the proof are corrected. A number of improvements was made thanks to the referee reports