Geometrical constructions of equilibrium states

TL;DR

This paper introduces geometric methods to analyze equilibrium states in partially hyperbolic systems, providing constructive proofs of key measures and detailed statistical properties, including applications to horocyclic flow.

Contribution

It offers a geometric approach to thermodynamic formalism for center isometries, including constructive proofs and new insights into equilibrium states and their properties.

Findings

Constructive proof of SRB and entropy maximizing measures.

Establishment of Bernoulliness and statistical properties.

Application to uniqueness of measures for horocyclic flow.

Abstract

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying in symbolic dynamics) and even provide new information in the classical case. For such systems, we give in particular a constructive proof of the existence of the SRB measure and of the entropy maximizing measure. It is also established very fine statistical properties (Bernoulliness), and it is given a characterization of equilibrium states in terms of their conditional measures in the stable/unstable lamination, similar to the SRB case. The construction is applied to obtain the uniqueness of quasi-invariant measures associated to H\"older Jacobian for the horocyclic flow.