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Equilibrium States for Partially Hyperbolic Maps with One-Dimensional Center

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We prove the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundle. We also prove, regarding a class of potentials, the uniqueness of such measures for endomorphisms defined on the 2-torus that: have a linear model as a factor; and with the condition that this measure gives zero weight to the set where the conjugacy with the linear model fails to be invertible. In particular, we obtain the uniqueness of the measure of maximal entropy. For the n-torus, the uniqueness in the case with one-dimensional center holds for absolutely partially hyperbolic maps with additional hypotheses on the invariant leaves, namely, dynamical coherence and quasi-isometry. We provide an example satisfying these hypotheses.
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Journal of Statistical Physics (2023) 190:194
https://doi.org/10.1007/s10955-023-03206-3
Equilibrium States for Partially Hyperbolic Maps with
One-Dimensional Center
Carlos F. Álvarez1
·Marisa Cantarino2
Received: 15 February 2023 / Accepted: 3 November 2023 / Published online: 25 November 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
We prove the existence of equilibrium states for partially hyperbolic endomorphisms with
one-dimensional center bundle. We also prove, regarding a class of potentials, the uniqueness
of such measures for endomorphisms defined on the 2-torus that: have a linear model as
a factor; and with the condition that this measure gives zero weight to the set where the
conjugacy with the linear model fails to be invertible. In particular, we obtain the uniqueness
of the measure of maximal entropy. For the n-torus, the uniqueness in the case with one-
dimensional center holds for absolutely partially hyperbolic maps with additional hypotheses
on the invariant leaves, namely, dynamical coherence and quasi-isometry. We provide an
example satisfying these hypotheses.
Keywords Equilibrium states ·Measures of maximal entropy ·Partially hyperbolic
endomorphisms ·Intrinsic ergodicity
Mathematics Subject Classification Primary: 37D35; Secondary: 37D30
Communicated by Peter Balint.
CF. A. was supported by ANID Proyecto FONDECYT postdoctorado 3230048, Chile. M. C. was partially
financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES), Fundação
Carlos Chagas Filho de Amparo à Pesquisa of the State of Rio de Janeiro (FAPERJ) and the Australian
Research Council (ARC).
BMarisa Cantarino
marisa.cantarino@monash.edu
Carlos F. Álvarez
carlos.alvarez.e@pucv.cl
1Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro
Barón, Valparaíso, Chile
2School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia
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