Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity

Suzete Maria Afonso; Vanessa Ramos; Jaqueline Siqueira

doi:10.3934/dcds.2021045

Article Contents

2021, Volume 41, Issue 9: 4484-4513. Doi: 10.3934/dcds.2021045

This issue Previous Article Centralizers of partially hyperbolic diffeomorphisms in dimension 3 Next Article

Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity

1.
Universidade Estadual Paulista (UNESP), Instituto de Geociências e Ciências Exatas, Câmpus de Rio Claro, Avenida 24-A, 1515, Bela Vista, Rio Claro, São Paulo, 13506-900, Brazil
2.
Centro de Ciências Exatas e Tecnologia - UFMA, Av. dos Portugueses, 1966, Bacanga, 65080-805, São Luís, Brazil
3.
*Instituto de Matemática - Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Cidade Universitária, Ilha do Fundão, Rio de Janeiro 21945-909, Brazil

^* Corresponding author: jaqueline@im.ufrj.br
^* Corresponding author: jaqueline@im.ufrj.br

Received: June 30, 2020

Revised: November 30, 2020

Early access: March 2021

Published: September 2021

The authors were supported by grant 2017/08732-1, São Paulo Research Foundation (FAPESP). VR was supported by grant Universal-01309/17, FAPEMA-Brazil. JS was supported by grant Universal-430154/2018-6, CNPq-Brazil

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Abstract

We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of Hölder continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity with respect to the potential of the equilibrium state as well as that of other thermodynamic quantities. Furthermore, we derive similar results for the equilibrium state associated to a family of non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials.

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Mathematics Subject Classification: Primary: 37A05, 37A30; Secondary: 37A35, 37D35.

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