Ruelle operator for continuous potentials and DLR-Gibbs measures

Leandro Cioletti; Artur O. Lopes; Manuel Stadlbauer

doi:10.3934/dcds.2020195

Article Contents

2020, Volume 40, Issue 8: 4625-4652. Doi: 10.3934/dcds.2020195

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Ruelle operator for continuous potentials and DLR-Gibbs measures

1.
Departamento de Matemática - UnB, 70910-900, Brasília, Brazil
2.
Departamento de Matemática - UFRGS, 91509-900, Porto Alegre, Brazil
3.
Departamento de Matemática - UFRJ, 21941-909, Rio de Janeiro, Brazil

^* Corresponding author: Leandro Cioletti
^* Corresponding author: Leandro Cioletti

Received: January 2019

Revised: October 2019

Published: May 2020

The authors would like to acknowledge financial support by CAPES, CNPq (PQ 313217/2018-1, PQ 407129/2013-8, PQ 312632/2018-5, Universal 426814/2016-9) and FAP-DF.

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Abstract

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability.

Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.

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Mathematics Subject Classification: Primary: 37D35, 28Dxx; Secondary: 37C30.

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References

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