Local rigidity for hyperbolic toral automorphisms
Mathematics Research Reports, Volume 3 (2022), pp. 57-68.

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 .

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DOI: 10.5802/mrr.13
Classification: 37D20, 37C15
Keywords: hyperbolic toral automorphism, conjugacy, local rigidity, linear cocycle
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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