Local rigidity for hyperbolic toral automorphisms

Boris Kalinin; Victoria Sadovskaya; Zhenqi Jenny Wang

doi:10.5802/mrr.13

Boris Kalinin ¹; Victoria Sadovskaya ¹; Zhenqi Jenny Wang 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, Volume 3 (2022), pp. 57-68.

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

Received: 2022-06-29
Revised: 2022-07-25
Published online: 2022-10-10

DOI: 10.5802/mrr.13

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

Author's affiliations:

Boris Kalinin ¹; Victoria Sadovskaya ¹; Zhenqi Jenny Wang 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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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

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