Rigidity of stable Lyapunov exponents and integrability for Anosov maps

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.