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Geometrical constructions of equilibrium states
Pablo D. Carrasco1; Federico Rodriguez-Hertz2
1 ICEx-UFMG, Av. Antônio Carlos, 6627 CEP 31270-901, Belo Horizonte, Brazil
2 Penn State, 227 McAllister Building, University Park, State College, PA16802
Mathematics Research Reports, Volume 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.

Received:
Revised:
Published online:
DOI: 10.5802/mrr.9
Classification: 37D35, 37D30
Keywords: equilibrium states, conditional measures, unique ergodicity
Pablo D. Carrasco 1; Federico Rodriguez-Hertz 2

1 ICEx-UFMG, Av. Antônio Carlos, 6627 CEP 31270-901, Belo Horizonte, Brazil
2 Penn State, 227 McAllister Building, University Park, State College, PA16802
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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