  In this paper, we give a complete topological and smooth classification of
non-invertible Anosov maps on torus. We show that two non-invertible Anosov
maps on torus are topologically conjugate if and only if their corresponding
periodic points have the same Lyapunov exponents on the stable bundles. As a
corollary, if two $C^r$ non-invertible Anosov maps on torus are topologically
conjugate, then the conjugacy is $C^r$-smooth along the stable foliation.
Moreover, we show that the smooth conjugacy class of a non-invertible Anosov
map on torus is completely determined by the Jacobians of return maps at
periodic points.

Dynamical Systems