Local rigidity for hyperbolic toral automorphisms

TL;DR

This paper investigates conditions under which conjugacies between hyperbolic toral automorphisms and their perturbations are smooth, providing new results that establish higher regularity of the conjugacy under certain weak differentiability assumptions.

Contribution

The authors prove that weak differentiability of the conjugacy implies it is $C^{1+\text{H"older}}$, and further show that if the automorphism is weakly irreducible, the conjugacy is actually smooth.

Findings

Weak differentiability of the conjugacy implies $C^{1+\text{H"older}}$ regularity.

Under weak irreducibility, the conjugacy is $C^{\infty}$.

Conditions for smoothness of conjugacy include matching periodic data and Lyapunov exponents.

Abstract

We consider a hyperbolic toral automorphism $L$ and its $C^1$-small perturbation $f$. It is well-known that $f$ is Anosov and topologically conjugate to $L$, but a conjugacy $H$ is only H\"older continuous in general. We discuss conditions for smoothness of $H$, such as conjugacy of the periodic data of $f$ and $L$, coincidence of their Lyapunov exponents, and weaker regularity of $H$, and we summarize questions, results, and techniques in this area. Then we introduce our new results: if $H$ is weakly differentiable then it is $C^{1+\text{H\"older}}$ and, if $L$ is also weakly irreducible, then $H$ is $C^\infty$.