SpringerLink
Log in

Pressure and Equilibrium States in Ergodic Theory

  • Reference work entry
  • First Online:
Ergodic Theory

Part of the book series: Encyclopedia of Complexity and Systems Science Series ((ECSSS))

  • 851 Accesses

  • Originally published in
  • R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, © Springer-Verlag 2009

Glossary

Dynamical system:

In this entry: a continuous transformation T of a compact metric space X. For each x ∈ X, the transformation T generates a trajectory (x,  Tx,  T2x, …).

Entropy:

In this entry: the maximal rate of information gain per time that can be achieved by coarse-grained observations on a measure-preserving dynamical system. This quantity is often denoted h(μ).

Equilibrium state:

In general, a given dynamical system T : X → X admits a huge number of invariant measures. Given some continuous ϕ : X → ℝ (“potential”), those invariant measures that maximize a functional of the form F(μ) = h(μ) + 〈ϕ, μ〉 are called “equilibrium states” for ϕ.

Ergodic theory:

Ergodic theory is the mathematical theory of measure-preserving dynamical systems.

Gibbs state:

In many cases, equilibrium states have a local structure that is determined by the local properties of the potential ϕ. They are called “Gibbs states.”

Invariant measure:

In this entry: a probability measure μ on Xwhich is...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 299.59
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
EUR 299.59
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  • Baladi V (2000) Positive transfer operators and decay of correlations. Advanced series in nonlinear dynamics, vol 16. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Bowen R (1970) Markov partitions for Axiom A diffeomorphisms. Am J Math 92:725–747

    Article  MathSciNet  MATH  Google Scholar 

  • Bowen R (1974a/1975) Some systems with unique equilibrium states. Math Syst Theory 8:193–202

    Google Scholar 

  • Bowen R (1974b/75) Bernoulli equilibrium states for Axiom A diffeomorphisms. Math Syst Theory 8:289–294

    Google Scholar 

  • Bowen R (1979) Hausdorff dimension of quasicircles. Inst Hautes Études Sci Publ Math 50:11–25

    Article  MathSciNet  MATH  Google Scholar 

  • Bowen R (2017) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture notes in mathematics, vol 470, 2nd edn (1st edn 1975). Springer, Berlin

    Google Scholar 

  • Bowen R, Ruelle D (1975) The ergodic theory of Axiom A flows. Invent Math 29:181–202

    Article  MathSciNet  MATH  Google Scholar 

  • Brudno AA (1983) Entropy and the complexity of the trajectories of a dynamical system. Trans Mosc Math Soc 2:127–151

    MATH  Google Scholar 

  • Bruin H, Keller G (1998) Equilibrium states for S-unimodal maps. Ergod Theory Dyn Syst 18(4):765–789

    Article  MathSciNet  MATH  Google Scholar 

  • Bruin H, Todd M (2008) Equilibrium states for potentials with sup φ −  inf φ < htop(f). Commun Math Phys 283(3):579–611. https://doi.org/10.1007/s00220-0-008-0596-0

    Article  Google Scholar 

  • Bruin H, Todd M (2009) Equilibrium states for interval maps: the potential −t log |Df|. Ann Sci Éc Norm Sup (4) 42(4):559–600. https://doi.org/10.24033/asens.2103

    Article  MATH  Google Scholar 

  • Bruin H, Demers M, Melbourne I (2010) Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod Theory Dyn Syst 30(3):687–728. https://doi.org/10.1017/S0143385709000200

    Article  MathSciNet  MATH  Google Scholar 

  • Buzzi J, Sarig O (2003) Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod Theory Dyn Syst 23(5):1383–1400

    Article  MathSciNet  MATH  Google Scholar 

  • Chaitin GJ (1987) Information, randomness & incompleteness. Papers on algorithmic information theory. World Scientific series in computer science, vol 8. World Scientific, Singapore

    MATH  Google Scholar 

  • Chernov N (2002) Invariant measures for hyperbolic dynamical systems. In: Handbook of dynamical systems, vol 1A. North-Holland, pp 321–407

    MATH  Google Scholar 

  • Coelho Z, Parry W (1990) Central limit asymptotics for shifts of finite type. Israel J Math 69(2):235–249

    Article  MathSciNet  MATH  Google Scholar 

  • Dobrushin RL (1968a) The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab Appl 13:197–224

    Article  MathSciNet  MATH  Google Scholar 

  • Dobrushin RL (1968b) Gibbsian random fields for lattice systems with pairwise interactions. Funct Anal Appl 2:292–301

    Article  MathSciNet  MATH  Google Scholar 

  • Dobrushin RL (1968c) The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct Anal Appl 2:302–312

    Article  MathSciNet  MATH  Google Scholar 

  • Dobrushin RL (1969) Gibbsian random fields. The general case. Funct Anal Appl 3:22–28

    Article  MathSciNet  MATH  Google Scholar 

  • Eizenberg A, Kifer Y, Weiss B (1994) Large deviations for ℤd-actions. Commun Math Phys 164(3):433–454

    Article  MathSciNet  MATH  Google Scholar 

  • Falconer K (2003) Fractal geometry. Mathematical foundations and applications, 2nd edn. Wiley, San Francisco

    Book  MATH  Google Scholar 

  • Fiebig D, Fiebig U-R, Yuri M (2002) Pressure and equilibrium states for countable state Markov shifts. Israel J Math 131:221–257

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher ME (1967) The theory of condensation and the critical point. Physics 3:255–283

    Article  MathSciNet  Google Scholar 

  • Fisher ME, Felderhof BU (1970) Phase transition in one-dimensional clusterinteraction fluids: IA. Thermodynamics, IB. Critical behavior. II. Simple logarithmic model. Ann Phys 58:177–280

    Google Scholar 

  • Gallavotti G (1996) Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. J Stat Phys 84:899–925

    Article  MathSciNet  MATH  Google Scholar 

  • Gallavotti G, Cohen EGD (1995) Dynamical ensembles in stationary states. J Stat Phys 80:931–970

    Article  MathSciNet  MATH  Google Scholar 

  • Gaspard P, Wang X-J (1988) Sporadicity: between periodic and chaotic dynamical behaviors. Proc Natl Acad Sci USA 85:4591–4595

    Article  MathSciNet  MATH  Google Scholar 

  • Georgii H-O (1988) Gibbs measures and phase transitions. In: de Gruyter studies in mathematics. de Gruyter, Berlin, p 9

    Google Scholar 

  • Gurevich BM, Savchenko SV (1998) Thermodynamic formalism for symbolic Markov chains with a countable number of states. Russ Math Surv 53(2):245–344

    Article  MathSciNet  MATH  Google Scholar 

  • Hofbauer F (1977) Examples for the nonuniqueness of the equilibrium state. Trans Am Math Soc 228:223–241

    Article  MathSciNet  MATH  Google Scholar 

  • Israel R (1979) Convexity in the theory of lattice gases. Princeton series in physics. Princeton University Press

    Google Scholar 

  • Jakobson M, Świątek (2002) One-dimensional maps. In: Handbook of dynamical systems, vol 1A. North-Holland, Amsterdam, pp 321–407

    Google Scholar 

  • Jaynes ET (1989) Papers on probability, statistics and statistical physics. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Jiang D, Qian M, Qian M-P (2000) Entropy production and information gain in Axiom A systems. Commun Math Phys 214:389–409

    Article  MathSciNet  MATH  Google Scholar 

  • Katok A (2007) Fifty years of entropy in dynamics: 1958–2007. J Mod Dyn 1(4):545–596

    Article  MathSciNet  MATH  Google Scholar 

  • Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Encyclopaedia of mathematics and its applications, vol 54. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Keller G (1998) Equilibrium states in ergodic theory. In: London Mathematical Society student texts, vol 42. Cambridge University Press, Cambridge

    Google Scholar 

  • Kifer Y (1990) Large deviations in dynamical systems and stochastic processes. Trans Am Math Soc 321:505–524

    Article  MathSciNet  MATH  Google Scholar 

  • Kolmogorov AN (1983) Combinatorial foundations of information theory and the calculus of probabilities. Uspekhi Mat Nauk 38:27–36

    MathSciNet  MATH  Google Scholar 

  • Lanford OE, Ruelle D (1969) Observables at infinity and states with short range correlations in statistical mechanics. Commun Math Phys 13:194–215

    Article  MathSciNet  Google Scholar 

  • Lewis JT, Pfister C-E (1995) Thermodynamic probability theory: some aspects of large deviations. Russ Math Surv 50:279–317

    Article  MathSciNet  MATH  Google Scholar 

  • Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Misiurewicz M (1976) A short proof of the variational principle for a \( {\mathrm{\mathbb{Z}}}_{+}^N \) action on a compact space. In: International conference on dynamical systems in mathematical physics (Rennes, 1975), Astérisque, No. 40, Soc Math France, pp 147–157

    Google Scholar 

  • Moulin-Ollagnier J (1985) Ergodic theory and statistical mechanics. In: Lecture notes in mathematics, vol 1115. Springer, Berlin

    MATH  Google Scholar 

  • Pesin Y (1997) Dimension theory in dynamical systems. Contemporary views and applications. University of Chicago Press, Chicago

    Book  MATH  Google Scholar 

  • Pesin Y, Senti S (2008) Equilibrium measures for maps with inducing schemes. J Mod Dyn 2(3):397–430. https://doi.org/10.3934/jmd.2008.2.397

    Article  MathSciNet  MATH  Google Scholar 

  • Pesin Y, Weiss (1997) The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1):89–106

    Article  MathSciNet  MATH  Google Scholar 

  • Pollicott M (2000) Rates of mixing for potentials of summable variation. Trans Am Math Soc 352(2):843–853

    Article  MathSciNet  MATH  Google Scholar 

  • Pomeau Y, Manneville P (1980) Intermittent transition to turbulence in dissipative dynamical systems. Commun Math Phys 74(2):189–197

    Article  MathSciNet  Google Scholar 

  • Rondoni L, Mejía-Monasterio C (2007) Fluctuations in nonequilibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity 20(10):R1–R37

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (1968) Statistical mechanics of a one-dimensional lattice gas. Commun Math Phys 9:267–278

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (1973) Statistical mechanics on a compact set with ℤν action satisfying expansiveness and specification. Trans Am Math Soc 185:237–251

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (1976) A measure associated with Axiom A attractors. Am J Math 98:619–654

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (1982) Repellers for real analytic maps. Ergod Theory Dyn Syst 2(1):99–107

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (1996) Positivity of entropy production in nonequilibrium statistical mechanics. J Stat Phys 85:1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (1998) Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J Stat Phys 95:393–468

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (2003) Extending the definition of entropy to nonequilibrium steady states. Proc Nat Acad Sci USA 100(6):3054–3058

    Article  MathSciNet  MATH  Google Scholar 

  • Ruelle D (2004) Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics, Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Sarig O (1999) Thermodynamic formalism for countable Markov shifts. Ergod Theory Dyn Syst 19(6):1565–1593

    Article  MathSciNet  MATH  Google Scholar 

  • Sarig O (2001) Phase transitions for countable Markov shifts. Commun Math Phys 217(3):555–577

    Article  MathSciNet  MATH  Google Scholar 

  • Sarig O (2003) Existence of Gibbs measures for countable Markov shifts. Proc Am Math Soc 131(6):1751–1758

    Article  MathSciNet  MATH  Google Scholar 

  • Seneta E (2006) Non-negative matrices and Markov chains. In: Springer series in statistics. Springer

    Google Scholar 

  • Sinai JG (1968) Markov partitions and C-diffeomorphisms. Funct Anal Appl 2:61–82

    Article  MathSciNet  MATH  Google Scholar 

  • Sinai JG (1972) Gibbs measures in ergodic theory. Russ Math Surv 27(4):21–69

    Article  MathSciNet  MATH  Google Scholar 

  • Thaler M (1980) Estimates of the invariant densities of endomorphisms with indifferent fixed points. Israel J Math 37(4):303–314

    Article  MathSciNet  MATH  Google Scholar 

  • Walters P (1975) Ruelle’s operator theorem and g-measures. Trans Am Math Soc 214:375–387

    MathSciNet  MATH  Google Scholar 

  • Walters P (1992) Differentiability properties of the pressure of a continuous transformation on a compact metric space. J Lond Math Soc (2) 46(3):471–481

    Google Scholar 

  • Wang X-J (1989) Statistical physics of temporal intermittency. Phys Rev A 40(11):6647–6661

    Article  Google Scholar 

  • Young L-S (1990) Large deviations in dynamical systems. Trans Am Math Soc 318:525–543

    MathSciNet  MATH  Google Scholar 

  • Young L-S (2002) What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J Stat Phys 108(5–6):733–754

    Article  Google Scholar 

  • Zinsmeister M (2000) Thermodynamic formalism and holomorphic dynamical systems. SMF/AMS texts and monographs, vol 2. American Mathematical Society

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Physique Théorique, CNRS/IP Paris, Palaiseau, France

    Jean-René Chazottes

  2. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Gerhard Keller

Authors
  1. Jean-René Chazottes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerhard Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-René Chazottes .

Editor information

Editors and Affiliations

  1. Mathematics, Williams College, Williamstown, MA, USA

    Cesar E. Silva

  2. B.Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv, Ukraine

    Alexandre I. Danilenko

Rights and permissions

Reprints and permissions

Copyright information

© 2023 Springer Science+Business Media, LLC, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Chazottes, JR., Keller, G. (2023). Pressure and Equilibrium States in Ergodic Theory. In: Silva, C.E., Danilenko, A.I. (eds) Ergodic Theory. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-2388-6_414

Download citation

Publish with us

Policies and ethics

Search

Navigation