Communications in Mathematical Physics

, Volume 292, Issue 3, pp 637–666 | Cite as

Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts



We find a necessary and sufficient condition for the Ruelle operator of a weakly Hölder continuous potential on a topologically mixing countable Markov shift to act with spectral gap on some rich Banach space. We show that the set of potentials satisfying this condition is open and dense for a variety of topologies. We then analyze the complement of this set (in a finer topology) and show that among the three known obstructions to spectral gap (weak positive recurrence, null recurrence, transience), transience is open and dense, and null recurrence and weak positive recurrence have empty interior.


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  1. ADU.
    Aaronson J., Denker M., Urbanski F.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337(2), 495–548 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. A.
    Aaronson J.: An introduction to infinite ergodic theory. Math. Surv. and Monog. 50, Providence. Amer. Math. Soc., Providence, RI (1997)Google Scholar
  3. AD.
    Aaronson J., Denker M.: Local limit theorems for Gibbs–Markov maps. Stochastics Dyn. 1, 193–237 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. B.
    Baladi V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16. Singapore, World Scientific (2000)Google Scholar
  5. BS.
    Buzzi J., Sarig O.: Uniqueness of equilibrium measures for countable Markov shifts and multi-dimensional piecewise expanding maps. Erg. Thy. Dynam. Sys. 23, 1383–1400 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. DF.
    Doeblin W., Fortet R.: Sur des chaînes à liasions complètes. Bull. Soc. Math. France 65, 132–148 (1937)MathSciNetGoogle Scholar
  7. GM.
    Gallavotti G., Miracle-Sole S.: Statistical mechanics of lattice systems. Commun. Math. Phys. 5(5), 317–323 (1967)MATHCrossRefMathSciNetADSGoogle Scholar
  8. G1.
    Gouëzel S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory Rel. Fields 128(1), 82–122 (2004)MATHCrossRefGoogle Scholar
  9. G2.
    Gouëzel S.: Regularity of coboundaries for nonuniformly expanding Markov maps. Proc. Amer. Math. Soc. 134(2), 391–401 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. GH.
    Guivarc’h Y., Hardy J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. (In French, English summary) [Limit theorems for a class of Markov chains and applications to Anosov diffeomorphisms] Ann. Inst. H. Poincaré Probab. Statist. 24(1), 73–98 (1988)MATHMathSciNetGoogle Scholar
  11. GS.
    Gurevich, B.M., Savchenko, S.V.: Thermodynamics formalism for countable Markov chains. Usp Mat. Nauk 53, 2, 3–106 (1998). Engl. transl. in Russ. Math. Surv. 53:2 3–106 (1998)Google Scholar
  12. HH.
    Hennion H., Hervé L.: Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi–compactness, LNM 1766. Springer, Berlin Heidelberg-New York (2001)CrossRefGoogle Scholar
  13. K.
    Kato, T.: Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Berlin: Springer-Verlag, 1995Google Scholar
  14. L.
    Ledrappier F.: Principe variationnel et systemes dynamiques symboliques. Z. Wahrs. Verb. Geb. 30, 185–202 (1974)MATHCrossRefMathSciNetGoogle Scholar
  15. Li.
    Liverani, C.: Central limit theorem for deterministic systems. International Conference on Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser. 362, Harlow: Longman, 1996, pp. 56–75Google Scholar
  16. Lo.
    Lopes A.: The zeta function, nondifferentiability of pressure, and the critical exponent of transition. Adv. Math. 101(2), 133–165 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. P.
    Parry W.: Entropy and generators in ergodic theory. W.A. Benjamin Inc., New York (1969)MATHGoogle Scholar
  18. PP.
    Parry, W.: Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–8 (1990)Google Scholar
  19. PrS.
    Prellberg T., Slawny J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2), 503–514 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  20. R.
    Ruelle D.: Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics. Second Edition. Cambridge Univ. Press, Cambridge (2004)Google Scholar
  21. Rt.
    Ruette S.: On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains. Pacific J. Math. 209(2), 365–380 (2003)MathSciNetCrossRefGoogle Scholar
  22. S1.
    Sarig O.: Thermodynamic formalism for countable Markov shifts. Erg. Th. Dynam. Syst. 19, 1565–1593 (1999)MATHCrossRefMathSciNetGoogle Scholar
  23. S2.
    Sarig O.: Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121, 285–311 (2001)MATHCrossRefMathSciNetGoogle Scholar
  24. S3.
    Sarig O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217, 555–577 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  25. S4.
    Sarig O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002)MATHCrossRefMathSciNetGoogle Scholar
  26. S5.
    Sarig O.: Critical exponents for dynamical systems. Commun. Math. Phys. 267, 631–667 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  27. VJ.
    Vere–Jones D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford 13(2), 7–28 (1962)MATHCrossRefMathSciNetGoogle Scholar
  28. Y.
    Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(2), 585–650 (1998)MATHCrossRefGoogle Scholar

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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