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Consider a topologically transitive countable Markov shift and, let $f$ be asummable potential with bounded variation and finite Gurevic pressure. We provethat there exists an equilibrium state $\mu_{tf}$ for each $t > 1$ and thatthere 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 ofthe family $(\mu_{tf})_{t>1}$. These results do not depend on the existence ofGibbs 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 zerotemperature limit beyond the finitely primitive case.

Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts