×

Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with Markovian structure. (English) Zbl 1484.37046

The authors consider a generalization of shifts of finite type for which the symbol space is replaced by a metric space \(X\) that is a finite union of clopen sets, and for each clopen set, there is a closed set of allowable follower symbols. They then equip the metric space with a probability measure. Given a Hölder continuous function on \(X\), they define an analogue of the Ruelle-Perron-Frobenius operator and prove results such as the existence of left and right eigenvectors for the operator, exponential convergence to the dominant eigenvector, etc.
They also apply their results to some special countable state shifts of finite type (for which the set of symbols that can follow a symbol \(i\), \(F(i):=\{j\colon i\to j\}\), takes finitely many values as \(i\) ranges over the alphabet). They provide a class of potentials for these systems, to which the Ruelle-type theorem mentioned above applies.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37B10 Symbolic dynamics
37B51 Multidimensional shifts of finite type
37A50 Dynamical systems and their relations with probability theory and stochastic processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aaronson, J.: An introduction to infinite ergodic theory, volume 50 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, (1997). doi:10.1090/surv/050 · Zbl 0882.28013
[2] Baraviera, A., Leplaideur, R., Lopes, A.: Ergodic optimization, zero temperature limits and the max-plus algebra. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, (2013). 29o Colóquio Brasileiro de Matemática. [29th Brazilian Mathematics Colloquium] · Zbl 1385.37011
[3] Baraviera, A.; Lopes, AO; Thieullen, P., A large deviation principle for the equilibrium states of Hölder potentials: the zero temperature case, Stoch. Dyn., 6, 1, 77-96 (2006) · Zbl 1088.60091 · doi:10.1142/S0219493706001657
[4] Baraviera, AT; Cioletti, LM; Lopes, AO; Mohr, J.; Souza, RR, On the general one-dimensional \(XY\) model: positive and zero temperature, selection and non-selection, Rev. Math. Phys., 23, 10, 1063-1113 (2011) · Zbl 1362.37022 · doi:10.1142/S0129055X11004527
[5] Bowen, R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (1975), Cham: Springer, Cham · Zbl 0308.28010 · doi:10.1007/BFb0081279
[6] Brémont, J., Gibbs measures at temperature zero, Nonlinearity, 16, 2, 419-426 (2003) · Zbl 1034.37007 · doi:10.1088/0951-7715/16/2/303
[7] Chauta, J., Freire, R.: Peierls barrier for countable markov shifts. (2019). arXiv:1904.09655
[8] Chazottes, J-R; Gambaudo, J-M; Ugalde, E., Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials, Ergodic Theory Dyn. Syst., 31, 4, 1109-1161 (2011) · Zbl 1230.37018 · doi:10.1017/S014338571000026X
[9] Cioletti, L.; Denker, M.; Lopes, AO; Stadlbauer, M., Spectral properties of the Ruelle operator for product-type potentials on shift spaces, J. Lond. Math. Soc., 95, 2, 684-704 (2017) · Zbl 1439.37025 · doi:10.1112/jlms.12031
[10] Cioletti, L., Correlation inequalities and monotonicity properties of the ruelle operator, Stoch. Dyn, 19, 6, 1950048 (2019) · Zbl 1453.37028 · doi:10.1142/S0219493719500485
[11] Cioletti, L.; Silva, EA; Stadlbauer, M., Thermodynamic formalism for topological markov chains on standard borel spaces, Discrete Contin. Dyn. Syst., 39, 11, 6277-6298 (2019) · Zbl 1436.37045 · doi:10.3934/dcds.2019274
[12] Contreras, G., Lopes, A.O., Oliveira, E.R.: Ergodic transport theory, periodic maximizing probabilities and the twist condition. In Modeling, dynamics, optimization and bioeconomics. I, volume 73 of Springer Proc. Math. Stat., pages 183-219. Springer, Cham, (2014). doi:10.1007/978-3-319-04849-9_12
[13] da Silva, EA; da Silva, RR; Souza, RRA, The analyticity of a generalized Ruelle‘s operator, Bull. Braz. Math. Soc., 45, 1, 53-72 (2014) · Zbl 1302.37008 · doi:10.1007/s00574-014-0040-3
[14] Freire, R.; Vargas, V., Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts, Trans. Am. Math. Soc., 370, 12, 8451-8465 (2018) · Zbl 1401.28018 · doi:10.1090/tran/7291
[15] Iommi, G., Ergodic optimization for renewal type shifts, Monatsh. Math., 150, 2, 91-95 (2007) · Zbl 1192.37015 · doi:10.1007/s00605-005-0389-x
[16] Jenkinson, O.; Mauldin, RD; Urbański, M., Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type, J. Stat. Phys., 119, 3-4, 765-776 (2005) · Zbl 1170.37307 · doi:10.1007/s10955-005-3035-z
[17] Kempton, T., Zero temperature limits of Gibbs equilibrium states for countable Markov shifts, J. Stat. Phys., 143, 4, 795-806 (2011) · Zbl 1219.82045 · doi:10.1007/s10955-011-0195-x
[18] Kitchens, B.P.: Symbolic dynamics. Universitext. Springer-Verlag, Berlin, (1998). One-sided, two-sided and countable state Markov shifts · Zbl 0892.58020
[19] Leplaideur, R., A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18, 6, 2847-2880 (2005) · Zbl 1125.37018 · doi:10.1088/0951-7715/18/6/023
[20] Leplaideur, R.; Watbled, F., Curie-weiss type models for general spin spaces and quadratic pressure in ergodic theory, J. Stat. Phys. (2020) · Zbl 1452.37009 · doi:10.1007/s10955-020-02579-z
[21] Leplaideur, R., Watbled, F.: Generalized curie-weiss-potts model and quadratic pressure in ergodic theory. (2020). arXiv:2003.09535 · Zbl 1452.37009
[22] Lopes, AO; Mengue, JK; Mohr, J.; Souza, RR, Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: positive and zero temperature, Ergodic Theory Dyn. Syst., 35, 6, 1925-1961 (2015) · Zbl 1352.37090 · doi:10.1017/etds.2014.15
[23] Lopes, A.O., Messaoudi, A., Stadlbauer, M., Vargas, V.: Invariant probabilities for discrete time linear dynamics via thermodynamic formalism. (2019). arXiv:1910.04902 · Zbl 1484.37044
[24] Lopes, AO; Mohr, J.; Souza, RR; Thieullen, P., Negative entropy, zero temperature and Markov chains on the interval, Bull. Braz. Math. Soc., 40, 1, 1-52 (2009) · Zbl 1181.60114 · doi:10.1007/s00574-009-0001-4
[25] Lopes, AO; Vargas, V., Gibbs states and gibbsian specifications on the space \({\mathbb{R}}^{{\mathbb{N}}} \), Dyn. Syst. Int. J., 35, 2, 216-241 (2020) · Zbl 1447.37046 · doi:10.1080/14689367.2019.1663789
[26] Lopes, AO; Vargas, V., The ruelle operator for symmetric \(\beta \)-shifts, Publ. Math., 64, 2, 661-680 (2020) · Zbl 1459.37029 · doi:10.5565/PUBLMAT6422012
[27] Mauldin, RD; Urbański, M., Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math., 125, 93-130 (2001) · Zbl 1016.37005 · doi:10.1007/BF02773377
[28] Mauldin, R.D., Urbański, M.: Graph directed Markov systems, volume 148 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, (2003). Geometry and dynamics of limit sets · Zbl 1033.37025
[29] Mohr, J.: Product type potential on the xy model: selection of maximizing probability and a large deviation principle. (2019). arXiv:1805.09858
[30] Parry, W.; Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 268, 187-188 (1990) · Zbl 0726.58003
[31] Ruelle, D., Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9, 267-278 (1968) · Zbl 0165.29102 · doi:10.1007/BF01654281
[32] Ruelle, D.: Thermodynamic formalism, volume 5 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., (1978). The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota · Zbl 0401.28016
[33] Sarig, O., Existence of Gibbs measures for countable Markov shifts, Proc. Am. Math. Soc., 131, 6, 1751-1758 (2003) · Zbl 1009.37003 · doi:10.1090/S0002-9939-03-06927-2
[34] Sarig, OM, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dyn. Syst., 19, 6, 1565-1593 (1999) · Zbl 0994.37005 · doi:10.1017/S0143385799146820
[35] Sinaĭ, Y.G.: Theory of phase transitions: rigorous results, volume 108 of International Series in Natural Philosophy. Pergamon Press, Oxford-Elmsford, N.Y., (1982). Translated from the Russian by J. Fritz, A. Krámli, P. Major and D. Szász · Zbl 0537.60097
[36] Thompson, CJ, Infinite-spin Ising model in one dimension, J. Math. Phys., 9, 241-245 (1968) · Zbl 1229.82053 · doi:10.1063/1.1664574
[37] van Enter, ACD; Ruszel, WM, Chaotic temperature dependence at zero temperature, J. Stat. Phys., 127, 3, 567-573 (2007) · Zbl 1147.82324 · doi:10.1007/s10955-006-9260-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.