  Given a 0-1 infinite matrix $A$ and its countable Markov shift $\Sigma_A$,
one of the authors and M. Laca have introduced a kind of {\it generalized
countable Markov shift} $X_A=\Sigma_A \cup Y_A$, where $Y_A$ is a special set
of finite admissible words. For some of the most studied countable Markov
shifts $\Sigma_A$, $X_A$ is a compactification of $\Sigma_A$, 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 $\Sigma_A$. New
phenomena appear, such as new conformal measures and a {\it length-type phase
transition}: the eigenmeasure lives on $\Sigma_A$ 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 $\Sigma_A$. 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.
