Smooth local rigidity for hyperbolic toral automorphisms

Abstract

We study the regularity of a conjugacy H H between a hyperbolic toral automorphism A A and its smooth perturbation f f . We show that if H H is weakly differentiable then it is C 1 + H\"older C^{1+\text {H\"older}} and, if A A is also weakly irreducible, then H H is C ∞ 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 ∞ C^\infty in prior local rigidity results.