Lyapunov exponents everywhere and rigidity

TL;DR

This paper investigates the conditions under which Anosov diffeomorphisms have Lyapunov exponents defined everywhere, leading to rigidity results that imply local structural stability for certain automorphisms of the torus.

Contribution

It establishes that having all Lyapunov exponents defined everywhere implies local rigidity of specific Anosov automorphisms on higher-dimensional tori.

Findings

All Lyapunov exponents defined everywhere implies local rigidity.

Rigidity results apply to $C^1$-close automorphisms of the torus.

Conditions involve irreducibility of the characteristic polynomial over $Q$.

Abstract

In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus $\mathbb{T}^d, d \geq 3,$ $C^1-$close to a linear automorphism diagonalizable over $\mathbb{R}$ and such that its characteristic polynomial is irreducible over $\mathbb{Q}.$