Lyapunov exponents everywhere and rigidity

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 ${\mathbb{T}}^d,d\ge 3,$ $C^1-$close to a linear automorphism diagonalizable over $\mathbb{R}$ and such that its characteristic polynomial is irreducible over $\mathbb{Q}\mathrm{.}$