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

Journal article

Pablo D. Carrasco, Federico Rodriguez Hertz
Israel Journal of Mathematics, 2023

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

Chicago/Turabian Click to copy
Carrasco, Pablo D., and Federico Rodriguez Hertz. “Contributions to the Ergodic Theory of Hyperbolic Flows: Unique Ergodicity for Quasi-Invariant Measures and Equilibrium States for the Time-One Map.” Israel Journal of Mathematics (2023).

MLA Click to copy
Carrasco, Pablo D., and Federico Rodriguez Hertz. “Contributions to the Ergodic Theory of Hyperbolic Flows: Unique Ergodicity for Quasi-Invariant Measures and Equilibrium States for the Time-One Map.” Israel Journal of Mathematics, 2023, doi:10.1007/s11856-023-2588-3.

BibTeX Click to copy

@article{carrasco2023a,
  title = {Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map},
  year = {2023},
  journal = {Israel Journal of Mathematics},
  doi = {10.1007/s11856-023-2588-3},
  author = {Carrasco, Pablo D. and Rodriguez Hertz, Federico}
}

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.