Smooth local rigidity for hyperbolic toral automorphisms

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 $C^{1+H\ddot{o}lder}$ 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 Hölder 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.

43 pages