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On the notions of singular domination and (multi-)singular hyperbolicity

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Abstract

The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincaré flow rather than the tangent flow in order to study the properties of the derivative. In this paper, we define the notion of singular domination, an analog of the dominated splitting for the linear Poincaré flow which is robust under perturbations. Based on this, we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti and da Luz (2017). The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows.

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Acknowledgements

The first author was supported by the European Research Council (Grant No. 692925). The third author was supported by National Natural Science Foundation of China (Grant Nos. 11671288, 11822109 and 11790274). The fourth author was supported by the starting grant from Beihang University and the European Research Council (Grant No. 692925). The authors thank Christian Bonatti, Lan Wen, Shaobo Gan and Yi Shi for helpful comments and discussions. This paper was prepared during the stay of the fourth author at Université Paris-sud, and he thanks Université Paris-sud for its hospitality. The authors are also grateful to the referees whose comments allowed to improve the presentation of the text.

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Sud, Orsay, 91405, France

    Sylvain Crovisier

  2. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Adriana da Luz

  3. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Dawei Yang

  4. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Jinhua Zhang

Authors
  1. Sylvain Crovisier
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  2. Adriana da Luz
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  3. Dawei Yang
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  4. Jinhua Zhang
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Corresponding author

Correspondence to Sylvain Crovisier.

Additional information

To the Memory of Professor Shantao Liao

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Crovisier, S., da Luz, A., Yang, D. et al. On the notions of singular domination and (multi-)singular hyperbolicity. Sci. China Math. 63, 1721–1744 (2020). https://doi.org/10.1007/s11425-019-1764-x

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  • DOI: https://doi.org/10.1007/s11425-019-1764-x

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