Rigidity of stable Lyapunov exponents and integrability for Anosov maps

TL;DR

This paper establishes conditions under which Anosov maps on tori have integrable unstable bundles, linking Lyapunov exponents at periodic points to the smoothness of conjugacies, with results depending on the dimension of the stable bundle.

Contribution

It provides new rigidity results connecting Lyapunov exponents, integrability, and smooth conjugacy for non-invertible Anosov maps on tori.

Findings

Stable bundle one-dimensional case: integrability characterized by uniform Lyapunov exponents at periodic points.

Higher-dimensional stable bundle case: similar results hold under $C^1$-perturbations of linear maps with simple spectra.

Topological conjugacy to linearization implies smooth conjugacy on the stable bundle.

Abstract

Let $f$ be a non-invertible irreducible Anosov map on $d$-torus. We show that if the stable bundle of $f$ is one-dimensional, then $f$ has the integrable unstable bundle, if and only if, every periodic point of $f$ admits the same Lyapunov exponent on the stable bundle with its linearization. For higher-dimensional stable bundle case, we get the same result on the assumption that $f$ is a $C^1$-perturbation of a linear Anosov map with real simple Lyapunov spectrum on the stable bundle. In both cases, this implies if $f$ is topologically conjugate to its linearization, then the conjugacy is smooth on the stable bundle.