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.
Similar content being viewed by others
References
J. Aaronson,Rational ergodicity and a metric invariant for Markov shifts, Israel Journal of Mathematics27 (1977), 93–123.
J. Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs50, American Mathematical Society, Providence R.I., 1997.
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.
L. M. Abramov,Entropy of induced automorphisms, Doklady Akademii Nauk SSSR128 (1959), 647–650.
R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics470 Springer-Verlag, Berlin, 1975.
W. Feller,An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn., Wiley, New York, 1968.
B. M. Gurevic,Topological entropy for denumerable Markov chains, Doklady Akademii Nauk SSSR187 (1969); English transl. in Soviet Mathematics Doklady10 (1969), 911–915.
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.
S. Isola,Dynamical zeta functions for non-uniformly hyperbolic transformations, preprint, 1997.
B. P. Kitchens,Symbolic Dynamics: One Sided, Two Sided and Countable State Markov Shifts, Universitext, Springer-Verlag, Berlin, 1998.
U. Krengel,Entropy of conservative transformations, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete7 (1967), 161–181.
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.
F. Ledrappier,Principe variationnel et systemes dynamique symbolique, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete30 (1974), 185–202.
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.
D. Ruelle,Thermodynamic Formalism, inEncyclopedia of Mathematics and its Applications, Vol. 5, Addison-Wesley, Reading, MA, 1978.
I. A. Salama,Topological entropy and recurrence of countable chains, Pacific Journal of Mathematics134 (1988), 325–341.
O. M. Sarig,Thermodynamic formalism for some countable topological Markov shifts, M.Sc. Thesis, Tel Aviv University, 1996.
O. M. Sarig,Thermodynamic formalism for countable Markov shifts, Ergodic Theory and Dynamical Systems19 (1999), 1565–1593.
S. V. Savchenko,Absence of equilibrium measure for nonrecurrent Holder functions, preprint (to appear in Matematicheskie Zametki).
E. Seneta,Non-negative Matrices and Markov Chains, Springer-Verlag, Berlin, 1973.
M. Thaler,Transformations on [0, 1] with infinite invariant measures, Israel Journal of Mathematics46 (1983), 67–96.
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.
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.
D. Vere-Jones,Geometric ergodicity in denumerable Markov chains, The Quarterly Journal of Mathematics. Oxford (2)13 (1962), 7–28.
D. Vere-Jones,Ergodic properties of nonnegative matrices—I, Pacific Journal of Mathematics22 (1967), 361–385.
P. Walters,Ruelle’s operator theorem and g-measures, Transactions of the American Mathematical Society214 (1975), 375–387.
P. Walters,Invariant measures and equilibrium states for some mappings which expand distances, Transactions of the American Mathematical Society236 (1978), 121–153.
M. Yuri,Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indagationes Mathematicae6 (1995), 355–383.
M. Yuri,Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory and Dynamical Systems19 (1999), 1365–1378.
M. Yuri,On the convergence to equilibrium states for certain non-hyperbolic systems, Ergodic Theory and Dynamical Systems17 (1997), 977–1000.
M. Yuri,Weak Gibbs measures for certain nonhyperbolic systems, preprint.
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02802508