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

Author(s)

Freire, Ricardo ^[1] ; Vargas, Victor ^{[1, 2]}

Affiliation

^[1] Univ Sao Paulo, IME, Dept Math, Rua Matao 1010, Sao Paulo - Brazil

^[2] Antonio Narino Univ, Fac Educ, Cl 22 Sur 12D-81, Bogota - Colombia

• Document type: Journal article
• Source: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY; v. 370, n. 12, p. 8451-8465, DEC 2018.
• Web of Science Citations: 2

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 -> 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. (AU)

• FAPESP's process: 11/16265-8 - Low dimensional dynamics
• Grantee: Edson Vargas
• Support Opportunities: Research Projects - Thematic Grants