Phase transitions in one-dimensional subshifts

Previous Article
Two problems related to prescribed curvature measures
DCDS Home
This Issue
Next Article
Actions of Baumslag-Solitar groups on surfaces

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

Abstract
Related Papers

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 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.
[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

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

[1]	Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[2]	Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[3]	Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[4]	Vítor Araújo. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 371-386. doi: 10.3934/dcds.2007.17.371
[5]	Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
[6]	Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks and Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257
[7]	Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001
[8]	Honghu Liu. Phase transitions of a phase field model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883
[9]	Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545
[10]	Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139
[11]	Christopher Hoffman. Subshifts of finite type which have completely positive entropy. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1497-1516. doi: 10.3934/dcds.2011.29.1497
[12]	Silvère Gangloff. Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 931-988. doi: 10.3934/dcds.2021143
[13]	Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593
[14]	Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
[15]	Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005
[16]	Steffen Arnrich. Modelling phase transitions via Young measures. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29
[17]	Paola Goatin. Traffic flow models with phase transitions on road networks. Networks and Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287
[18]	Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163
[19]	Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[20]	Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Nicolai T. A. Haydn
on Google Scholar
Nicolai T. A. Haydn

[Back to Top]