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.

Received:

Revised:

Published online:

DOI:
10.5802/mrr.9

Revised:

Published online:

Classification:
37D35, 37D30

Keywords: equilibrium states, conditional measures, unique ergodicity

Keywords: equilibrium states, conditional measures, unique ergodicity

Author's affiliations:

Pablo D. Carrasco ^{1};
Federico Rodriguez-Hertz ^{2}

License: CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{MRR_2021__2__45_0, author = {Pablo D. Carrasco and Federico Rodriguez-Hertz}, title = {Geometrical constructions of equilibrium states}, journal = {Mathematics Research Reports}, pages = {45--54}, publisher = {MathOA foundation}, volume = {2}, year = {2021}, doi = {10.5802/mrr.9}, language = {en}, url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.9/} }

TY - JOUR AU - Pablo D. Carrasco AU - Federico Rodriguez-Hertz TI - Geometrical constructions of equilibrium states JO - Mathematics Research Reports PY - 2021 SP - 45 EP - 54 VL - 2 PB - MathOA foundation UR - https://mrr.centre-mersenne.org/articles/10.5802/mrr.9/ DO - 10.5802/mrr.9 LA - en ID - MRR_2021__2__45_0 ER -

Pablo D. Carrasco; Federico Rodriguez-Hertz. Geometrical constructions of equilibrium states. Mathematics Research Reports, Volume 2 (2021), pp. 45-54. doi : 10.5802/mrr.9. https://mrr.centre-mersenne.org/articles/10.5802/mrr.9/

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