Smooth Conjugacy of Anosov Diffeomorphisms on Higher Dimensional Tori

Let L be a hyperbolic automorphism of T, d ≥ 3. We study the smooth conjugacy problem in a small C-neighborhood U of L. The main result establishes C regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are C-close to an irreducible linear hyperbolic automorphism L with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on T with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb C conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of T C-close to the original example.