Local rigidity for Anosov automorphisms

TL;DR

This paper proves that certain irreducible Anosov automorphisms of tori are locally rigid under small perturbations with the same periodic data, and shows these automorphisms are generic in SL(d,Z).

Contribution

It establishes local rigidity for a class of Anosov automorphisms with specific eigenvalue conditions, extending understanding of their stability under perturbations.

Findings

Anosov automorphisms with no three eigenvalues sharing the same modulus are locally rigid.

Such automorphisms are generic in SL(d,Z).

Eigenvalue assumptions are crucial, as shown by constructed examples.

Abstract

We consider an irreducible Anosov automorphism L of a torus T^d such that no three eigenvalues have the same modulus. We show that L is locally rigid, that is, L is C^1 conjugate to any C^1-small perturbation f with the same periodic data. We also prove that toral automorphisms satisfying these assumptions are generic in SL(d,Z). Examples constructed in the Appendix by Rafael de la Llave show importance of the assumption on the eigenvalues.