Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States

2018/08/02 by Rodrigo Bissacot, Bissacot, Rodrigo, Ruy Exel +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1808.00765

openalex publication_date 2018/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a 0-1 infinite matrix A and its countable Markov shift Σ_A, one of the authors and M. Laca have introduced a kind of \it generalized countable Markov shift X_A=Σ_A ∪ Y_A, where Y_A is a special set of finite admissible words. For some of the most studied countable Markov shifts Σ_A, X_A is a compactification of Σ_A, and always it is at least locally compact. We developed the thermodynamic formalism on the space X_A, exploring the connections with standard results on Σ_A. New phenomena appear, such as new conformal measures and a \it length-type phase transition: the eigenmeasure lives on Σ_A at high temperature and lives on Y_A at low temperature. Using a pressure-point definition proposed by M. Denker and M. Yuri for iterated function systems, we proved that the Gurevich pressure is a natural definition for the pressure function in the generalized setting. For the gauge action, the Gurevich entropy is a critical temperature for the existence of new conformal measures (KMS states) living on Y_A. We exhibit examples with infinitely (even uncountable) many new extremal conformal measures, undetectable in the usual formalism. We prove that conformal measures always exist at low temperatures when the potential is coercive enough. We characterized a basis of the topology of X_A to study the weak^* convergence of measures on X_A, and we show some cases where the conformal measure living on Y_A converges to a conformal one living on Σ_A. We prove the equivalence among several notions of conformality for locally compact Hausdorff second countable spaces, including quasi-invariant measures for generalized Renault-Deaconu groupoids.

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• Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts

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