Israel Journal of Mathematics

, Volume 131, Issue 1, pp 221–257 | Cite as

Pressure and equilibrium states for countable state markov shifts

Article

Abstract

We give a general definition of the topological pressurePtop(f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if εf<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]
    R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics470, Springer-Verlag, Berlin, 1975.MATHGoogle Scholar
  2. [G]
    B. M. Gurevic,Topological entropy of enumerable Markov Chains, Doklady Akademii Nauk SSSR187 (1969); English transl.: Soviet Mathematics Doklady10 (1969), 911–915.Google Scholar
  3. [GS]
    B. M. Gurevic and S. V. Savchenko,Thermodynamic formalism for countable symbolic Markov chains, Russian Mathematical Survey53 (1998), 245–344.CrossRefGoogle Scholar
  4. [K]
    B. Kitchens,Symbolic Dynamics, Springer, Berlin, 1998.MATHGoogle Scholar
  5. [P]
    K. R. Parthasaraty,Probability Measures on Metric Spaces, Academic Press, New York, 1967.Google Scholar
  6. [PP]
    Ya. Pesin and B. Pitskel,Topological pressure and the variational principle for non-compact sets, Functional Analysis and its Applications18 (1984), 307–318.MATHCrossRefMathSciNetGoogle Scholar
  7. [S1]
    O. Sarig,Thermodynamic formalism for countable state Markov shifts, Ergodic Theory and Dynamical Systems19 (1999), 1565–1593.MATHCrossRefMathSciNetGoogle Scholar
  8. [S2]
    O. Sarig,Thermodynamic formalism for null recurrent potentials, Israel Journal of Mathematics121 (2001), 285–312.MATHMathSciNetGoogle Scholar
  9. [S3]
    O. Sarig,Thermodynamic formalism for countable Markov shifts, Thesis, Tel Aviv University, April 2000.Google Scholar
  10. [R]
    D. Ruelle,Thermodynamic formalism, inEncyclopedia of Mathematics and its Applications, Vol. 5, Addison-Wesley, Reading, MA, 1978.Google Scholar
  11. [W1]
    P. Walters,An Introduction to Ergodic Theory, GTM, Springer-Verlag, Berlin, 1981.Google Scholar
  12. [W2]
    P. Walters,Invariant measures and equilibrium states for some mappings which expand distances, Transactions of the American Mathematical Society236 (1978), 121–153.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2002

Authors and Affiliations

  1. 1.Institut für Mathematische StochastikUniversität GöttingenGöttingenGermany
  2. 2.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany
  3. 3.Department of Business AdministrationSapporo University NishiokaToyohira-ku, SapporoJapan

Personalised recommendations