  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.
