Hostname: page-component-84b7d79bbc-5lx2p Total loading time: 0 Render date: 2024-07-27T11:59:13.058Z Has data issue: false hasContentIssue false
  • English
  • Français

On the relation between Gibbs and $g$-measures

Published online by Cambridge University Press:  28 March 2018

STEVEN BERGHOUT Open the ORCID record for STEVEN BERGHOUT [Opens in a new window] ,
ROBERTO FERNÁNDEZ  and
EVGENY VERBITSKIY
STEVEN BERGHOUT
Affiliation:
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands email s.berghout@math.leidenuniv.nl
ROBERTO FERNÁNDEZ
Affiliation:
Department of Mathematics, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands email R.Fernandez1@uu.nl
EVGENY VERBITSKIY
Affiliation:
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands email s.berghout@math.leidenuniv.nl Department of Mathematics, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands email evgeny@math.leidenuniv.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

Thermodynamic formalism, the theory of equilibrium states, is studied both in dynamical systems and probability theory. Various closely related notions have been developed: e.g. Dobrushin–Lanford–Ruelle Gibbs, Bowen–Gibbs and $g$-measures. We discuss the relation between Gibbs and $g$-measures in a one-dimensional context. Often $g$-measures are also Gibbs, but recently an example to the contrary has been presented. In this paper we discuss exactly when a $g$-measure is Gibbs and how this relates to notions such as uniqueness and reversibility of $g$-measures.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3224 - 3249
Copyright
© Cambridge University Press, 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bissacot, R., Endo, E., van enter, A. and Le Ny, A.. Entropic repulsion and lack of the $g$ -measure property for Dyson models. Preprint, 2017, arXiv:1705.03156.Google Scholar
Bousch, T.. La condition de Walters. Ann. Sci. Eć. Norm. Supér. (4) 34 (2001), 287–311.Google Scholar
Bousch, T. and Jenkinson, O.. Cohomology classes of dynamically non-negative C k functions. Invent. Math. 148(1) (2002), 207–217.Google Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Systems Theory 8(3) (1974/75), 193–202.Google Scholar
Coelho, Z. and Quas, A. N.. Criteria for d -continuity. Trans. Amer. Math. Soc. 350(8) (1998), 3257–3268.Google Scholar
Dachian, S. and Nahapetian, B. S.. Description of specifications by means of probability distributions in small volumes under condition of very weak positivity. J. Stat. Phys. 117(1–2) (2004), 281–300.Google Scholar
Dinaburg, E. I.. A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190 (1970), 19–22 (in Russian).Google Scholar
Dobrushin, R. L.. Description of a random field by means of conditional probabilities and conditions for its regularity. Teor. Veroyatn. Primen. 13 (1968), 201–229 (in Russian, with English summary).Google Scholar
Dobrushin, R. L.. Gibbsian random fields for lattice systems with pairwise interactions. Funktsional. Anal. i Prilozhen. 2(4) (1968), 31–43 (in Russian).Google Scholar
Dobrushin, R. L.. Lecture Given at the Workshop Probability and Physics. Renkum, Holland, 1995.Google Scholar
Dobrushin, R. L. and Shlosman, S. B.. Gibbsian description of ‘non-Gibbsian’ fields. Russian Math. Surveys 52(2) (1997), 285–297.Google Scholar
Fernández, R. and Maillard, G.. Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9(6) (2004), 145–176 (electronic).Google Scholar
Fernández, R. and Maillard, G.. Chains with complete connections: general theory, uniqueness, loss of memory and mixing properties. J. Stat. Phys. 118(3–4) (2005), 555–588.Google Scholar
Fernández, R., Gallo, S. and Maillard, G.. Regular g-measures are not always Gibbsian. Electron. Commun. Probab. 16 (2011), 732–740.Google Scholar
Georgii, H.-O.. Stochastische Felder und ihre Anwendung auf Interaktionssysteme (Lecture Notes) . Inst. für Angewandte Math., 1974.Google Scholar
Goodman, T. N. T.. Relating topological entropy and measure entropy. Bull. Lond. Math. Soc. 3 (1971), 176–180.Google Scholar
Hulse, P.. An example of non-unique g-measures. Ergod. Th. & Dynam. Sys. 26(2) (2006), 439–445.Google Scholar
Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.Google Scholar
Kozlov, O. K.. A Gibbs description of a system of random variables. Probl. Inf. Transm. 10(3) (1974), 94–103 (in Russian).Google Scholar
Lanford, O. E.. Statistical Mechanics and Mathematical Problems. Ed. Lenard, A.. Springer, Berlin, 1973, pp. 1–113.Google Scholar
Lanford, O. E. III and Ruelle, D.. Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys. 13 (1969), 194–215.Google Scholar
Ledrappier, F.. Principe variationnel et systèmes dynamiques symboliques. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30 (1974), 185–202.Google Scholar
Palmer, M. R., Parry, W. and Walters, P.. Large sets of endomorphisms and of g-measures. The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) (Lecture Notes in Mathematics, 668) . Springer, Berlin, 1978, pp. 191–210.Google Scholar
Ruelle, D.. Thermodynamic formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics (Encyclopedia of Mathematics and its Applications, 5) . Addison-Wesley, Reading, MA, 1978, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota.Google Scholar
Sinaĭ, J. G.. Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27(4(166)) (1972), 21–64 (in Russian).Google Scholar
Sullivan, W. G.. Potentials for almost Markovian random fields. Comm. Math. Phys. 33 (1973), 61–74.Google Scholar
Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar
Walters, P.. Erratum: ‘A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function’ [Dyn. Syst. 18(2) (2003), 131–138; MR1994784]. Dyn. Syst. 18(3) (2003), 271–278.Google Scholar
Walters, P.. Regularity conditions and Bernoulli properties of equilibrium states and g-measures. J. Lond. Math. Soc. (2) 71(2) (2005), 379–396.Google Scholar
Walters, P.. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys. 27(4) (2007), 1323–1348.Google Scholar
Yuri, M.. Zeta functions for certain non-hyperbolic systems and topological Markov approximations. Ergod. Th. & Dynam. Sys. 18(6) (1998), 1589–1612.Google Scholar