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Rigidity of Stable Lyapunov Exponents and Integrability for Anosov Maps

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Abstract

Let f be a non-invertible irreducible Anosov map on d-torus. We show that if the stable bundle of f is one-dimensional, then f has the integrable unstable bundle, if and only if, every periodic point of f admits the same Lyapunov exponent on the stable bundle as its linearization. For higher-dimensional stable bundle case, we get the same result on the assumption that f is a \(C^1\)-perturbation of a linear Anosov map with real simple Lyapunov spectrum on the stable bundle. In both cases, this implies if f is topologically conjugate to its linearization, then the conjugacy is smooth on the stable bundle.

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Acknowledgements

S. Gan was partially supported by National Key R &D Program of China (2020YFE0204200) and NSFC (11831001, 12161141002). Y. Shi was partially supported by National Key R &D Program of China (2021YFA1001900) and NSFC (12071007, 11831001, 12090015).

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jinpeng An, Shaobo Gan & Ruihao Gu

  2. School of Mathematics, Sichuan University, Chengdu, 610065, China

    Yi Shi

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  1. Jinpeng An
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  2. Shaobo Gan
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  4. Yi Shi
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An, J., Gan, S., Gu, R. et al. Rigidity of Stable Lyapunov Exponents and Integrability for Anosov Maps. Commun. Math. Phys. 402, 2831–2877 (2023). https://doi.org/10.1007/s00220-023-04786-7

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