Rigidity and Absolute Continuity of Foliations of Anosov Endomorphisms

We found a dichotomy involving rigidity and measure of maximal entropy of a \(C^{\infty }\) $C^{\infty }$ -special Anosov endomorphism of the 2-torus. Considering \(\widetilde{m} \) $\tilde{m}$ the measure of maximal entropy of a \(C^{\infty }\) $C^{\infty }$ -special Anosov endomorphism of the 2-torus, either \(\widetilde{m}\) $\tilde{m}$ satisfies the Pesin formula (in this case we get smooth conjugacy with the linearization) or there is a set Z,  such that \(\widetilde{m}(Z) = 1,\) $\tilde{m}(Z)=1,$ but Z intersects every unstable leaf on a set of zero measure of the leaf. Also, we can characterize the absolute continuity of the intermediate foliation for a class of volume-preserving special Anosov endomorphisms of \(\mathbb {T}^3\) ${\mathbb{T}}^3$ .