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May  2013, 33(5): 1965-1973. doi: 10.3934/dcds.2013.33.1965

Phase transitions in one-dimensional subshifts

Nicolai T. A. Haydn 1,

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States

Received  December 2011 Revised  August 2012 Published  December 2012

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.
Keywords: phase transitions., equilibrium states, measure of maximal entropy, Subshifts.
Mathematics Subject Classification: Primary: 37D35; Secondary: 82B2.
Citation: Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1965-1973. doi: 10.3934/dcds.2013.33.1965
References:
[1]

R. Bowen, Some systems with unique equilibrium states, Math. Syst. Th., 8 (1975), 193-202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. O. Georgii, "Gibbs Measures and Phase Transitions," de Gruyter Studies in Mathematics, 9, 1988. doi: 10.1515/9783110850147.  Google Scholar

[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.  Google Scholar

[6]

Hofbauer, Examples for the nonuniqueness of the equilibrium state, AMS Transactions, 228 (1977), 223-241.  Google Scholar

[7]

W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104.  Google Scholar

[8]

P Walter, "An Introduction to Ergodic Theory,'' Springer-Verlag, GTM, 79, 1982  Google Scholar

show all references

References:
[1]

R. Bowen, Some systems with unique equilibrium states, Math. Syst. Th., 8 (1975), 193-202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. O. Georgii, "Gibbs Measures and Phase Transitions," de Gruyter Studies in Mathematics, 9, 1988. doi: 10.1515/9783110850147.  Google Scholar

[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.  Google Scholar

[6]

Hofbauer, Examples for the nonuniqueness of the equilibrium state, AMS Transactions, 228 (1977), 223-241.  Google Scholar

[7]

W. Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory, 8 (1974), 97-104.  Google Scholar

[8]

P Walter, "An Introduction to Ergodic Theory,'' Springer-Verlag, GTM, 79, 1982  Google Scholar

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