\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the work of Sarig on countable Markov chains and thermodynamic formalism (Brin Prize article)

Abstract Related Papers Cited by
  • The paper is a nontechnical survey and is aimed to illustrate Sarig'sprofound contributions to statistical physics and in particular,thermodynamic formalism for countable Markov shifts. I will discusssome of Sarig's work on characterization of existence of Gibbsmeasures, existence and uniqueness of equilibrium states as well asphase transitions for Markov shifts on a countable set of states.
    Mathematics Subject Classification: 37D40, 37D25, 37D40.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    J. Aaronson and M. Denker, Local limit theorems for Gibbs-Markov maps, Stochastics Dyn., 1 (2001), 193-237.doi: 10.1142/S0219493701000114.

    [2]

    J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rational maps, Trans. AMS, 337 (1993), 495-548.doi: 10.1090/S0002-9947-1993-1107025-2.

    [3]

    R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, Edited by J.-R. Chazottes, Lect. Notes Math., 470, Springer-Verlag, Berlin, 2008.

    [4]

    J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergod. Th. and Dyn. Syst., 23 (2003), 1383-1400.doi: 10.1017/S0143385703000087.

    [5]

    V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts, Comm. Math. Phys., 292 (2009), 637-666.doi: 10.1007/s00220-009-0891-4.

    [6]

    R. Dobrušin, Description of a random field by means of conditional probabilities and conditions for its regularity, Teor. Veroyatnoistei i Primenenia, 13 (1968), 201-229; English translation in Theory of Prob. and Appl., 13 (1968), 197-223.

    [7]

    R. Dobrušin, The problem of uniqueness of a Gibbsian random field and the problem of phase transitions, Functional Anal. Appl., 2 (1968), 302-312.

    [8]

    M. Gordin, On the Central Limit Theorem for stationary processes, (Russian) Doklady Akademii Nauk SSSR, 188 (1969), 739-741; English translation in Soviet Math. Dokl., 10 (1969), 1174-1176.

    [9]

    B. M. Gurevič, Topological entropy for denumerable Markov chains, Dokl. Acad. Nauk SSSR, 187 (1969), 715-718; English translation in Soviet Math. Dokl., 10 (1969), 911-915.

    [10]

    B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph, Dokl. Acad. Nauk SSSR, 192 (1970), 963-965; English translation in Soviet Math. Dokl., 11 (1970), 744-747.

    [11]

    B. M. Gurevič, A variational characterization of one-dimensional countable state Gibbs random fields, Z. Wahrsch. Verw. Gebiete, 68 (1984), 205-242.doi: 10.1007/BF00531778.

    [12]

    B. M. Gurevič and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chainswith a countable number of states, Uspehi. Mat. Nauk, 53 (1998), 3-106; English translation in Russian Math. Surv., 53 (1998), 245-344.doi: 10.1070/rm1998v053n02ABEH000017.

    [13]

    G. Keller, Equilibrium States in Ergodic Theory, Cambridge University Press, Cambridge, 1998.doi: 10.1017/CBO9781107359987.

    [14]

    O. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys., 13 (1969), 194-215.doi: 10.1007/BF01645487.

    [15]

    F. Ledrappier, On Omri Sarig's work on the dynamics on surfaces, J. Modern Dynamics, (2014).

    [16]

    R. Mauldin and M. Urbański, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math., 125 (2001), 93-130.doi: 10.1007/BF02773377.

    [17]

    W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66.doi: 10.1090/S0002-9947-1964-0161372-1.

    [18]

    D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278.doi: 10.1007/BF01654281.

    [19]

    D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.doi: 10.1017/CBO9780511617546.

    [20]

    D. Ruelle, A variational formulation of equilibrium statistical mechanics and the Gibbs state rule, Comm. Math. Phys., 5 (1967), 324-329.doi: 10.1007/BF01646446.

    [21]

    O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Theory and Dyn. Syst., 19 (1999), 1565-1593.doi: 10.1017/S0143385799146820.

    [22]

    O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311.doi: 10.1007/BF02802508.

    [23]

    O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577.doi: 10.1007/s002200100367.

    [24]

    O. Sarig, On an example with a non-analytic topological pressure, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 311-315.doi: 10.1016/S0764-4442(00)00189-0.

    [25]

    O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. of AMS, 131 (2003), 1751-1758.doi: 10.1090/S0002-9939-03-06927-2.

    [26]

    O. Sarig, Thermodynamic Formalism for Countable Markov Shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593.doi: 10.1017/S0143385799146820.

    [27]

    O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426.doi: 10.1090/S0894-0347-2012-00758-9.

    [28]

    Y. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation in Russan Math. Surveys, 27 (1972), 21-69.

    [29]

    Y. Sinai, Construction of Markov partitions, Functional Anal. and Appl., 2 (1968), 245-253.doi: 10.1007/BF01076126.

    [30]

    D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford Ser. (2), 13 (1962), 7-28.doi: 10.1093/qmath/13.1.7.

    [31]

    D. Vere-Jones, Ergodic properties of nonnegative matrices. I, Pac. J. Math., 22 (1967), 361-386.doi: 10.2140/pjm.1967.22.361.

    [32]

    P. Walter, Ruelle's operator theorem and $g$-measures, Trans. AMS, 214 (1975), 375-387.

    [33]

    P. Walter, Invariant measures and equilibrium states for some mappings which expand distances, Trans. AMS, 236 (1978), 121-153.doi: 10.1090/S0002-9947-1978-0466493-1.

    [34]

    P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

    [35]

    M. Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. (N. S.), 6 (1995), 355-383.doi: 10.1016/0019-3577(95)93202-L.

    [36]

    M. Yuri, On the convergence to equilibrium states for certain non-hyperbolic systems, Ergod. Theory and Dyn. Syst., 17 (1997), 977-1000.doi: 10.1017/S0143385797086240.

  • 加载中
Open Access Under a Creative Commons license
SHARE

Article Metrics

HTML views() PDF downloads(108) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return