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On the Regularity of Invariant Foliations

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Abstract

We show that the stable invariant foliation of codimension 1 near a zero-dimensional hyperbolic set of a \(C^{\beta}\) map with \(\beta>1\) is \(C^{1+\varepsilon}\) with some \(\varepsilon>0\). The result is applied to the restriction of higher regularity maps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.

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ACKNOWLEDGMENTS

I am grateful to Dongchen Li for useful discussions. I dedicate this paper to the 70th birthday of my dear friend Sergey Gonchenko. Our quest into the depths of the Newhouse domain that started more than 35 years ago from his interest in \(\Omega\)-moduli and led us to many breathtaking discoveries has determined the course of my scientific life.

Funding

This work was supported by the Leverhulme Trust grant RPG-2021-072 and by the RScF grant 19-71-10048.

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  1. Imperial College, SW7 2AZ, London, UK

    Dmitry Turaev

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Correspondence to Dmitry Turaev.

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MSC2010

37D10,37D05,37G25

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Turaev, D. On the Regularity of Invariant Foliations. Regul. Chaot. Dyn. 29, 6–24 (2024). https://doi.org/10.1134/S1560354724010027

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