Topological and smooth classification of Anosov maps on torus

TL;DR

This paper provides a comprehensive topological and smooth classification of non-invertible Anosov maps on the torus, linking conjugacy to Lyapunov exponents and Jacobians at periodic points.

Contribution

It establishes that topological conjugacy is characterized by Lyapunov exponents and Jacobians, and shows smoothness of conjugacy along stable foliations.

Findings

Topological conjugacy corresponds to equal Lyapunov exponents on stable bundles.

Smooth conjugacy along stable foliation is guaranteed for $C^r$ maps.

Conjugacy class is determined by Jacobians of return maps at periodic points.

Abstract

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.