EQUILIBRIUM STATES AND ZERO TEMPERATURE LIMIT ON TOPOLOGICALLY TRANSITIVE COUNTABLE MARKOV SHIFTS

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 -> infinity. We also prove that the Kolmogorov-Sinai entropy is continuous at infinity with respect to the parameter t, that is, lim(t -> 8) h(mu(tf)) = h((mu infinity)), where mu(infinity) 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.