Consider a topologically transitive countable Markov shift and, let $f$ be a summable potential with bounded variation and finite Gurevic pressure. We prove that there exists an equilibrium state $\mu_{ tf}$ for each $t > 1$ and that there exists accumulation points for the family $(\mu_{ tf})_{ t>1}$ as $t \to \infty$. We also prove that the Kolmogorov-Sinai entropy is continuous at $\infty$ with respect to the parameter $t$, that is $\lim_{ t \to \infty} h(\mu_{ tf})=h(\mu_{ \infty})$, where $\mu_{ \infty}$ is an accumulation point of the family $(\mu_{ tf})_{ t>1}$. These results do not depend on the existence of Gibbs measures and, therefore, they extend results of \cite{ MaUr01} and \cite{ Sar99} for the existence of equilibrium states without the BIP property, \cite{ JMU05} for the existence of accumulation points in this case and, finally, we extend completely the result of \cite{ Mor07} for the entropy zero temperature limit beyond the finitely primitive case.
Categories: math.DS math-ph math.MP
