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 & Continuous Dynamical Systems - A, 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.   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.  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.  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), 744.   Google Scholar

[6]

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

[7]

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

[8]

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

show all references

References:
[1]

R. Bowen, Some systems with unique equilibrium states,, Math. Syst. Th., 8 (1975), 193.   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.  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.  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), 744.   Google Scholar

[6]

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

[7]

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

[8]

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

[1]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

[2]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[3]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[4]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[5]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[6]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[8]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[9]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[10]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

[11]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[12]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]