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arXiv · 2305.17494

Centralizer classification and rigidity for some partially hyperbolic toral automorphisms

Sven Sandfeldt
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Abstract

In this paper we consider local centralizer classification and rigidity of some toral automorphisms. In low dimensions we classify up to finite index possible centralizers for volume preserving diffeomorphisms $f$ $C^{1}-$close to an ergodic irreducible toral automorphism $L$. Moreover, we show a rigidity result in the case that the centralizer of $f$ is large: If the smooth centralizer $Z^{\infty}(f)$ is virtually isomorphic to that of $L$ then $f$ is $C^{\infty}-$conjugate to $L$. In higher dimensions we show a similar rigidity result for certain irreducible toral automorphisms. We also classify up to finite index all possible centralizers for symplectic diffeomorphisms $C^{5}-$close to a class of irreducible symplectic automorphisms on tori of any dimension.

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Sven Sandfeldt. 2023-05-27. Centralizer classification and rigidity for some partially hyperbolic toral automorphisms. https://doi.org/10.3934/jmd.2024013

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