Article Contents
Article Contents

# Phase transitions in one-dimensional subshifts

• In this note we give simple examples of one-dimensional mixing subshift with positive topological entropy which have two distinct measures of maximal entropy. We also give examples of subshifts which have two mutually singular equilibrium states for Hölder continuous functions. We also indicate how the construction can be extended to yield examples with any number of equilibrium states.
Mathematics Subject Classification: Primary: 37D35; Secondary: 82B26.

 Citation:

•  [1] R. Bowen, Some systems with unique equilibrium states, Math. Syst. Th., 8 (1975), 193-202. [2] R. Burton and J. E. Steif, Nonuniqueness of measures of maximal entropy for subshifts of finite type, Ergod. Th. Dynam. Sys., 14 (1994), 213-235.doi: 10.1017/S0143385700007859. [3] R. Burton and J. E. Steif, New results on measures of maximal entropy, Israel J. Math., 89 (1995), 275-300.doi: 10.1007/BF02808205. [4] H. O. Georgii, "Gibbs Measures and Phase Transitions," de Gruyter Studies in Mathematics, 9, 1988.doi: 10.1515/9783110850147. [5] B. M. Gurevic, Shift entropy and Markov measures in the path space of a denumerable graph, Dokl. Akad. Nauk. SSSR, 192 (1970); Engl. transl. in Soviet Math. Dokl., 11 (1970), 744-747. [6] Hofbauer, Examples for the nonuniqueness of the equilibrium state, AMS Transactions, 228 (1977), 223-241. [7] W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104. [8] P Walter, "An Introduction to Ergodic Theory,'' Springer-Verlag, GTM, 79, 1982