Local rigidity of Lyapunov spectrum for toral automorphisms

TL;DR

This paper proves that small volume-preserving perturbations of certain toral automorphisms with matching Lyapunov exponents are smoothly conjugate to the original automorphism, demonstrating local rigidity of the Lyapunov spectrum.

Contribution

It establishes local rigidity results for conjugacies between toral automorphisms and their perturbations under specific eigenvalue and irreducibility conditions.

Findings

Perturbations with matching Lyapunov exponents are smoothly conjugate to the original automorphism.

Results extend to partially hyperbolic automorphisms with two-dimensional center bundles.

Conditions include eigenvalue restrictions and irreducibility over Q.

Abstract

We study the regularity of the conjugacy between an Anosov automorphism $L$ of a torus and its small perturbation. We assume that $L$ has no more than two eigenvalues of the same modulus and that $L^4$ is irreducible over $\mathbb Q$. We consider a volume-preserving $C^1$-small perturbation $f$ of $L$. We show that if Lyapunov exponents of $f$ with respect to the volume are the same as Lyapunov exponents of $L$, then $f$ is $C^{1+\text{H\"older}}$ conjugate to $L$. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.