Skip to main content

Advertisement

Log in

Thermodynamic formalism for null recurrent potentials

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We extend Ruelle’s Perron-Frobenius theorem to the case of Hölder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(ϕ) denote the Gurevic pressure of ϕ and letL ϕ be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatL *ϕ ν=e P(ϕ)ν,L ϕ h=e P(ϕ) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL kϕ , and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. Aaronson,Rational ergodicity and a metric invariant for Markov shifts, Israel Journal of Mathematics27 (1977), 93–123.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs50, American Mathematical Society, Providence R.I., 1997.

    MATH  Google Scholar 

  3. J. Aaronson, M. Denker and M. Urbanski,Ergodic theory for Markov fibered systems and parabolic rational maps, Transactions of the American Mathematical Society337 (1993), 495–548.

    Article  MATH  MathSciNet  Google Scholar 

  4. L. M. Abramov,Entropy of induced automorphisms, Doklady Akademii Nauk SSSR128 (1959), 647–650.

    MATH  MathSciNet  Google Scholar 

  5. R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics470 Springer-Verlag, Berlin, 1975.

    MATH  Google Scholar 

  6. W. Feller,An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn., Wiley, New York, 1968.

    MATH  Google Scholar 

  7. B. M. Gurevic,Topological entropy for denumerable Markov chains, Doklady Akademii Nauk SSSR187 (1969); English transl. in Soviet Mathematics Doklady10 (1969), 911–915.

  8. B. M. Gurevic,Shift entropy and Markov measures in the path space of a denumerable graph, Doklady Akademii Nauk SSSR192 (1970); English transl. in Soviet Mathematics Doklady11 (1970), 744–747.

  9. S. Isola,Dynamical zeta functions for non-uniformly hyperbolic transformations, preprint, 1997.

  10. B. P. Kitchens,Symbolic Dynamics: One Sided, Two Sided and Countable State Markov Shifts, Universitext, Springer-Verlag, Berlin, 1998.

    Google Scholar 

  11. U. Krengel,Entropy of conservative transformations, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete7 (1967), 161–181.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Lasota and Y. A. Yorke,On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society186 (1973), 481–488.

    Article  MathSciNet  Google Scholar 

  13. F. Ledrappier,Principe variationnel et systemes dynamique symbolique, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete30 (1974), 185–202.

    Article  MATH  MathSciNet  Google Scholar 

  14. V. A. Rohlin,Exact endomorphisms of a Lebesgue space, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya25 (1961), 499–530 (Russian); English Translation in American Mathematical Society Translations, Series 2,39 (1964), 1–37.

    MathSciNet  Google Scholar 

  15. D. Ruelle,Thermodynamic Formalism, inEncyclopedia of Mathematics and its Applications, Vol. 5, Addison-Wesley, Reading, MA, 1978.

    Google Scholar 

  16. I. A. Salama,Topological entropy and recurrence of countable chains, Pacific Journal of Mathematics134 (1988), 325–341.

    MathSciNet  Google Scholar 

  17. O. M. Sarig,Thermodynamic formalism for some countable topological Markov shifts, M.Sc. Thesis, Tel Aviv University, 1996.

  18. O. M. Sarig,Thermodynamic formalism for countable Markov shifts, Ergodic Theory and Dynamical Systems19 (1999), 1565–1593.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. V. Savchenko,Absence of equilibrium measure for nonrecurrent Holder functions, preprint (to appear in Matematicheskie Zametki).

  20. E. Seneta,Non-negative Matrices and Markov Chains, Springer-Verlag, Berlin, 1973.

    Google Scholar 

  21. M. Thaler,Transformations on [0, 1] with infinite invariant measures, Israel Journal of Mathematics46 (1983), 67–96.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Thaler,A limit theorem for the Perron-Frobenius operator of transformations on [0, 1] with indifferent fixed points, Israel Journal of Mathematics91 (1998), 111–129.

    Article  MathSciNet  Google Scholar 

  23. M. Urbanski and P. Hanus,A new class of positive recurrent functions, inGeometry and Topology in Dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemporary Mathematics246, American Mathematical Society, 1999, pp. 123–135.

  24. D. Vere-Jones,Geometric ergodicity in denumerable Markov chains, The Quarterly Journal of Mathematics. Oxford (2)13 (1962), 7–28.

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Vere-Jones,Ergodic properties of nonnegative matrices—I, Pacific Journal of Mathematics22 (1967), 361–385.

    MATH  MathSciNet  Google Scholar 

  26. P. Walters,Ruelle’s operator theorem and g-measures, Transactions of the American Mathematical Society214 (1975), 375–387.

    Article  MATH  MathSciNet  Google Scholar 

  27. P. Walters,Invariant measures and equilibrium states for some mappings which expand distances, Transactions of the American Mathematical Society236 (1978), 121–153.

    Article  MathSciNet  Google Scholar 

  28. M. Yuri,Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indagationes Mathematicae6 (1995), 355–383.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Yuri,Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory and Dynamical Systems19 (1999), 1365–1378.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Yuri,On the convergence to equilibrium states for certain non-hyperbolic systems, Ergodic Theory and Dynamical Systems17 (1997), 977–1000.

    Article  MATH  MathSciNet  Google Scholar 

  31. M. Yuri,Weak Gibbs measures for certain nonhyperbolic systems, preprint.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarig, O.M. Thermodynamic formalism for null recurrent potentials. Isr. J. Math. 121, 285–311 (2001). https://doi.org/10.1007/BF02802508

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02802508

Keywords

Navigation