Thermodynamic formalism for generalized countable Markov shifts

Countable Markov shifts, denoted by $\Sigma_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $\Sigma_A$ in the locally compact case. The set $\Sigma_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal measure in it, and exploring its connections with the usual formalism for $\Sigma_A$. New phenomena appear, as different types of phase transitions and new conformal measures undetected by the classical thermodynamic formalism for $A$ not row-finite. Given a potential $F$ and inverse of temperature $\beta$, we study the existence of conformal measures $\mu_{\beta}$ associated to $\beta F$. We present examples where there exists a critical $\beta_c$ s. t. we have existence of conformal probabilities satisfying $\mu_{\beta}(\Sigma_A)=0$ for every $\beta > \beta_c$ and, on the weak$^*$ topology, the set of conformal probabilities for $\beta >\beta_c$ collapses to the standard conformal probability $\mu_{\beta_c}$, $\mu_{\beta_c}(\Sigma_A)=1$, for the limit $\beta\to\beta_c$. We study in detail the generalized renewal shift and modifications of it. We highlight the bijection between infinite emitters of the alphabet and extremal conformal probabilities 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; such measures are not detected on the standard renewal shift since for low temperatures, $\beta F$ is transient.