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arXiv · 2609.10769

Symplecto-rigidity and bootstrap for Anosov symplectomorphisms and contact Anosov flows

Andrey Gogolev·Federico Rodriguez Hertz
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Abstract

This paper is a sequel to the authors' paper on rigidity of higher dimensional contact Anosov flows~[GRH]. At the time the authors were unaware of much earlier work of Hamenst\"adt~[Ham] devoted to the same problem. The methods of~[GRH] and~[Ham] are overlapping but not entirely the same. In this paper we strengthen some the results of Hamenst\"adt, by further bootstrapping the regularity of the conjugacy of 2-pinched contact Anosov flows. We also introduce a notion of symplecto-rigidity for symplectomorphisms and prove that some Anosov diffeomorphisms are symplecto-rigid. We further combine symplecto-rigidity with various rigidity techniques to establish a number of smooth rigidity results of Anosov symplectomorphisms. Some new phenomena are present in the realm of Anosov symplectomorphism. For example, while the well-known de la Llave example on the 4-torus demonstrates absence of rigidity in the space of smooth Anosov diffeomorphisms, however, we prove that it is rigid in the space of Anosov sympelctomorphisms.

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Andrey Gogolev, Federico Rodriguez Hertz. 2026-09-09. Symplecto-rigidity and bootstrap for Anosov symplectomorphisms and contact Anosov flows. https://arxiv.org/abs/2609.10769

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