Geometrical constructions of equilibrium states

Carrasco, Pablo D.; Rodriguez-Hertz, Federico

Mathematics Research Reports, Tome 2 (2021), pp. 45-54.

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic systems—center isometries—that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying on 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. We also establish very fine statistical properties (Bernoulliness), and we give 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 Hölder Jacobians for the horocyclic flow.

Classification : 37D35, 37D30
Mots clés : equilibrium states, conditional measures, unique ergodicity

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