Local rigidity for hyperbolic toral automorphisms

Kalinin, Boris; Sadovskaya, Victoria; Wang, Zhenqi Jenny

doi:10.5802/mrr.13

Kalinin, Boris ¹ ; Sadovskaya, Victoria ¹ ; Wang, Zhenqi Jenny ²

¹ Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824,USA

Mathematics Research Reports, Tome 3 (2022), pp. 57-68.

Résumé

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

Reçu le : 2022-06-29
Révisé le : 2022-07-25
Publié le : 2022-10-10

DOI : 10.5802/mrr.13

Classification : 37D20, 37C15
Mots clés : hyperbolic toral automorphism, conjugacy, local rigidity, linear cocycle

Affiliations des auteurs :

Kalinin, Boris ¹ ; Sadovskaya, Victoria ¹ ; Wang, Zhenqi Jenny ²

¹ Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824,USA

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Kalinin, Boris; Sadovskaya, Victoria; Wang, Zhenqi Jenny. Local rigidity for hyperbolic toral automorphisms. Mathematics Research Reports, Tome 3 (2022), pp. 57-68. doi : 10.5802/mrr.13. http://archive.numdam.org/articles/10.5802/mrr.13/

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