Lyapunov Exponents Everywhere and Rigidity

Abstract

In the present work, we obtain rigidity results analyzing the set of regular points, in the sense of Oseledec’s Theorem. It is presented a study on the possibility of Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus \(\mathbb {T}^{d}, d \geq 3,\) C1 −close to a linear automorphism diagonalizable over \(\mathbb {R}\) and such that its characteristic polynomial is irreducible over \(\mathbb {Q}.\)