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Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

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Abstract

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with Hölder Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon measures on the horocyclic foliation such that any probability (invariant or not) having conditionals given by this family, necessarily is the unique equilibrium state of the system.

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Acknowledgments

The authors thank Barbara Schapira for bringing to our attention the references [1, 20, 19, 12]. We also thank the referee for her/his careful reading, and for pointing out some mistakes and omissions.

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Authors and Affiliations

  1. Institute de Ciências Exatas, Universidade Federal de Minas Gerais, Avda. Presidente Antonio Carlos 6627, Belo Horizonte-MG, BR31270-90, Brazil

    Pablo D. Carrasco

  2. Department of Mathematics, Pennsylvania State University, 227 McAllister Building, University Park, State College, PA, 16802, USA

    Federico Rodriguez-Hertz

Authors
  1. Pablo D. Carrasco
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  2. Federico Rodriguez-Hertz
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Corresponding author

Correspondence to Federico Rodriguez-Hertz.

Additional information

Partially supported by FAPEMIG Universal APQ-02160-21.

Partially supported by NSF grant DMS-1900778.

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Carrasco, P.D., Rodriguez-Hertz, F. Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2588-3

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  • DOI: https://doi.org/10.1007/s11856-023-2588-3

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