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

Abstract:

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 [Israel J. Math. 125 (2001), pp. 93–130] and [Ergodic Theory Dynam. Systems 19 (1999), pp. 1565–1593] for the existence of equilibrium states without the big images and preimages (BIP) property, [J. Stat. Phys. 119 (2005), pp. 765–776] for the existence of accumulation points in this case and, finally, we extend completely the result of [J. Stat. Phys. 126 (2007), pp. 315–324] for the entropy zero temperature limit beyond the finitely primitive case.