Thermodynamic formalism for generalized countable Markov shifts

• Autor(a) principal: Thiago Costa Raszeja
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://doi.org/10.11606/T.45.2020.tde-06012021-103444

Countable Markov shifts, which we denote by $\\Sigma_A$ for a 0-1 infinite matrix $A$, are central objects in Symbolic Dynamics and Ergodic Theory. R. Exel and M. Laca have introduced the corresponding operator algebras as a generalization of the Cuntz-Krieger algebras for an infinite and countable alphabet. They introduced the set $X_A=\\Sigma_A \\cup Y_A$, a kind of generalized countable Markov shift that coincides with the space $\\Sigma_A$ in the locally compact case. The space $X_A$ contains as dense subsets the standard countable Markov $\\Sigma_A$ and a subset of finite allowed words $Y_A$. The last one is dense when it is non-empty. We develop the thermodynamic formalism for the generalized countable Markov shifts $X_A$, introducing the notion of conformal measure in $X_A$ and exploring its connections with the usual thermodynamic formalism for $\\Sigma_A$. New phenomena appear, as different types of phase transitions and new conformal measures that are not detected in the classical thermodynamic formalism when the matrix $A$ is not row-finite. Given a potential $F$ and inverse of temperature $\\beta$, we study the problem of the existence and absence of conformal measures $\\mu_{\\beta}$ associated with $\\beta F$. We present examples where there exists a critical $\\beta_c$ such that we have the existence of conformal probabilities satisfying $\\mu_{\\beta}(\\Sigma_A)=0$ for every $\\beta > \\beta_c$ and, on the weak$^*$ topology, when we take the limit on $\\beta$ going to $\\beta_c$, the set of conformal probabilities for the inverse of temperature $\\beta > \\beta_c$ collapses to the standard conformal probability $\\mu_{\\beta_c}$ such that $\\mu_{\\beta_c}(\\Sigma_A)=1$. We study in detail the generalized renewal shift and modifications of it. We highlight the bijection between the elements of the alphabet, which are infinite emitters, and extremal conformal probability measures for this class of renewal type shifts. We prove the existence and uniqueness of the eigenmeasure (probability) of the Ruelle\'s transformation at low enough temperature for a particular potential on the generalized renewal shift; these measures are not detected on the standard renewal shift since for low temperatures, the potential $\\beta F$ is transient.