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Maximal Transverse Measures of Expanding Foliations

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Abstract

For an expanding (unstable) foliation of a diffeomorphism, we use a natural dynamical averaging to construct transverse measures, which we call maximal, describing the statistics of how the iterates of a given leaf intersect the cross-sections to the foliation. For a suitable class of diffeomorphisms, we prove that this averaging converges, even exponentially fast, and the limit measures have finite ergodic decompositions. These results are obtained through relating the maximal transverse measures to the maximal u-entropy measures of the diffeomorphism (see [UVYY]).

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Notes

  1. All transverse measures are taken to be not identically zero, unless stated otherwise.

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Acknowledgements

We are grateful to the anonymous reviewers for several comments and corrections which greatly helped us improve the presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong, China

    Raul Ures

  2. SUSTech International Center for Mathematics, Shenzhen, Guangdong, China

    Raul Ures

  3. IMPA, Est. D. Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Marcelo Viana

  4. Department of Mathematics, Wake Forest University, Winston-Salem, NC, USA

    Fan Yang

  5. Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil

    Jiagang Yang

Authors
  1. Raul Ures
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  2. Marcelo Viana
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  3. Fan Yang
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  4. Jiagang Yang
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Corresponding author

Correspondence to Marcelo Viana.

Additional information

Communicated by C. Liverani.

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R. U. was partially supported by NNSFC 11871262, NNSFC 12071202, and NNSFC 12161141002. M.V. and J.Y. were partially supported by CNPq, FAPERJ, and PRONEX. We acknowledge support from the Fondation Louis D–Institut de France (project coordinated by M. Viana).

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Ures, R., Viana, M., Yang, F. et al. Maximal Transverse Measures of Expanding Foliations. Commun. Math. Phys. 405, 121 (2024). https://doi.org/10.1007/s00220-024-04993-w

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