Local rigidity for hyperbolic toral automorphisms

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 Hölder 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 $C^{1+\text{Ho¨lder}}$ and, if $L$ is also weakly irreducible, then $H$ is $C^{\infty }$.

13 pages. arXiv admin note: text overlap with arXiv:2111.01309