Rigidity for Some Cases of Anosov Endomorphisms of Torus

We obtain smooth conjugacy between non-necessarily special Anosov endomorphisms in the conservative case. Among other results, we prove that a strongly special $C^{\mathrm{\infty }}-$Anosov endomorphism of ${\mathbb{T}}^2$ and its linearization are smoothly conjugated since they have the same periodic data. Assuming that for a strongly special $C^{\mathrm{\infty }}-$Anosov endomorphism of ${\mathbb{T}}^2$ every point is regular (in Oseledec’s Theorem sense), then we obtain again smooth conjugacy with its linearization. We also obtain some results on local rigidity of linear Anosov endomorphisms of $d-$torus, where $d\ge 3,$ under periodic data assumption. The study of differential equations defined on invariant leaves plays an important role in rigidity problems such as those treated here.