Smooth local rigidity for hyperbolic toral automorphisms

TL;DR

This paper proves that conjugacies between hyperbolic toral automorphisms and their smooth perturbations are smoothly regular under certain conditions, advancing local rigidity results in hyperbolic dynamics.

Contribution

It establishes that weakly differentiable conjugacies are actually $C^{1+H"older}$, and $C^ abla$ if the automorphism is weakly irreducible, improving regularity results.

Findings

Weakly differentiable conjugacies are $C^{1+H"older}$.

Under irreducibility, conjugacies are $C^ abla$.

Improves regularity of conjugacies in local rigidity theorems.

Abstract

We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its smooth perturbation $f$ We show that if $H$ is weakly differentiable then it is $C^{1+H\"older}$ and, if $A$ is also weakly irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish results of independent interest on H\"older continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to $C^\infty$ in prior local rigidity results.