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

Phase transitions in one-dimensional subshifts

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.
Citation: Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete and 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.

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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.

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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.

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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

show all references

References:
[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

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