Abstract
Let Σ A be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f:Σ A → IR, the family of Gibbs-equilibrium states (μ tf )t⩾1 for the functions tf is shown to be tight. Any weak*-accumulation point as t→∞ is shown to be a maximizing measure for f.
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References
M. Aizenman E.H. Lieb (1981) ArticleTitleThe third law of thermodynamics and the degeneracy of the ground state for lattice systems J Stat Phys. 24 279–297 Occurrence Handle10.1007/BF01007649
P. Billingsley (1968) Convergence of Probability Measures John Wiley & Sons New York
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lecture Notes in Mathematics Vol.~470 (Berlin-Heidelberg-New York, Springer-Verlag, 1975)
J. Brémont (2003) ArticleTitleOn the behaviour of Gibbs measures at temperature zero Nonlinearity 16 419–426 Occurrence Handle10.1088/0951-7715/16/2/303
J. Buzzi O. Sarig (2003) ArticleTitleUniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps Ergod. Th. Dyn. Syst. 23 1383–1400 Occurrence Handle10.1017/S0143385703000087
Z. N. Coelho, Entropy and ergodicity of skew-products over subshifts of finite type and central limit asymptotics, Ph.D. Thesis (Warwick University, 1990).
G. Contreras A.O. Lopes Ph. Thieullen (2001) ArticleTitleLyapunov minimizing measures for expanding maps of the circle Ergod. Th. Dyn. Syst. 21 1379–1409 Occurrence Handle10.1017/S0143385701001663
Conze J.-P. and Guivarc’h Y. (1993). Croissance des sommes ergodiques, manuscript. circa
O. Jenkinson (2001) ArticleTitleGeometric barycentres of invariant measures for circle maps Ergod. Th. Dyn. Syst. 21 511–532 Occurrence Handle10.1017/S0143385701001250
Jenkinson O., Mauldin R.D. and Urbański M. Ergodic optimization for countable state subshifts of finite type, preprint.
A. Kerimov (1998) ArticleTitleCountable extreme Gibbs states in a one-dimensional model with unique ground state J Phys A: Math. Gen. 31 2815–2821 Occurrence Handle10.1088/0305-4470/31/12/007 Occurrence Handle1:CAS:528:DyaK1cXisVGnsbw%3D
Mauldin R.D. and Urbański M. Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets (Cambridge University Press, 2003)
M. Pollicott R. Sharp (1994) ArticleTitleRates of recurrence for Z^qand Rq extensions of subshifts of finite type J. London Math. Soc. 49 401–416
C. Radin (1989) ArticleTitleOrdering in lattice gases at low temperature J. Phys. A 22 317–319
C. Radin (1991) ArticleTitleDisordered ground states of classical lattice models Rev. Math. Phys. 3 125–135 Occurrence Handle10.1142/S0129055X91000059
C. Radin, Zn versus Z actions for systems of finite type, Symbolic dynamics and its applications (New Haven, CT, 1991), 339–342, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.
D. Ruelle, Thermodynamic Formalism, Encyclopaedia of Mathematics and its Applications, Vol. 5 (Addison-Wesley, 1978)
O. Sarig (1999) ArticleTitleThermodynamic formalism for countable Markov shifts Ergod. Th. Dyn. Syst. 19 1565–1593 Occurrence Handle10.1017/S0143385799146820
O. Sarig (2003) ArticleTitleCharacterization of existence of Gibbs measures for countable Markov shifts Proc. Am. Math. Soc. 131 1751–1758
R. Schrader (1970) ArticleTitleGround states in classical lattice systems with hard core Comm. Math. Phys. 16 247–264 Occurrence Handle10.1007/BF01646534
P. Walters, An Introduction to Ergodic Theory (Springer, 1981).
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Jenkinson, O., Mauldin, R.D. & Urbański, M. Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type. J Stat Phys 119, 765–776 (2005). https://doi.org/10.1007/s10955-005-3035-z
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DOI: https://doi.org/10.1007/s10955-005-3035-z