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

TL;DR

This paper investigates the ergodic properties of hyperbolic flows, establishing unique ergodicity for certain measures and characterizing equilibrium states for the time-one map in Anosov systems.

Contribution

It proves the uniqueness of transverse quasi-invariant measures with Hölder Jacobians and characterizes equilibrium states via Radon measures on horocyclic foliations.

Findings

Uniqueness of transverse quasi-invariant measures with Hölder Jacobians.

Characterization of equilibrium states through Radon measures on horocyclic foliations.

Establishment of conditions under which probability measures are the unique equilibrium states.

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 H\"older 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.