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Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure

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Abstract

Consider a compact metric space \((M, d_M)\) and \(X = M^{{\mathbb {N}}}\). We prove a Ruelle’s Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from a continuous function \(A : M \times M \rightarrow {\mathbb {R}}\) that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle \(S^1\). Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.

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Acknowledgements

The authors are very grateful to the professor Artur Oscar Lopes for his unconditional support, helpful talks and extremely useful suggestions that improved the final version of this paper. The second author would to thank to PNPD-CAPES, INCTMat and the Francisco José de Caldas Fund by the financial support during part of the development of this paper.

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  1. Department of Mathematics, IME-UFRGS, Porto Alegre, 91509-900, Brazil

    Rafael Rigão Souza

  2. School of Mathematics, National University of Colombia, Medellín, 050034, Colombia

    Victor Vargas

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  1. Rafael Rigão Souza
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Souza, R.R., Vargas, V. Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure. Qual. Theory Dyn. Syst. 21, 5 (2022). https://doi.org/10.1007/s12346-021-00537-y

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