$C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus

Abstract

We consider two $C^2$ Anosov diffeomorphisms in a $C^1$ neighborhood of a linear hyperbolic automorphism ofthree dimensional torus with real spectrum. We prove that they are $C^{1+\nu}$ conjugate if and only if thedifferentials of the return maps at corresponding periodic points have the same eigenvalues.