# Lyapunov exponents and rigidity of Anosov automorphisms and skew products

## TL;DR

This paper establishes local rigidity results for linear Anosov diffeomorphisms and skew products, showing that small perturbations preserving certain Lyapunov exponents are smoothly conjugate or have smooth center foliations.

## Contribution

It proves that preserving Lyapunov exponents under small perturbations implies smooth conjugacy for linear Anosov automorphisms and smoothness of the center foliation for skew products.

## Key findings

- Perturbations preserving Lyapunov exponents are smoothly conjugate to the original automorphism.
- The center foliation remains smooth under volume-preserving perturbations with the same Lyapunov exponents.
- Rigidity results are extended to skew products over Anosov automorphisms.

## Abstract

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation $f$ of $f_0$ with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.