Advertisement

Communications in Mathematical Physics

, Volume 125, Issue 2, pp 239–262 | Cite as

The thermodynamic formalism for expanding maps

  • David Ruelle
Article

Abstract

Letf:XX be an expanding map of a compact space (small distances are increased by a factor >1). A generating functionζ(z) is defined which countsf-periodic points with a weight. One can expressζ in terms of nonstandard “Fredholm determinants” of certain “transfer operators”, which can be studied by methods borrowed from statistical mechanics. In this paper we review the spectral properties of the transfer operators and the corresponding analytic properties ofζ(z). Gibbs distributions and applications to Julia sets are also discussed. Some new results are proved, and some natural conjectures are proposed.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Statistical Mechanic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotone transformations. PreprintGoogle Scholar
  2. 2.
    Bowen, R.: Markov partitions for Axiom A diffeomorphisms. Trans. Am. Math. Soc.154, 377–397 (1971)Google Scholar
  3. 3.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol.470. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  4. 4.
    Bowen, R.: On Axiom A Diffeomorphisms. CBMS Regional Conf. Series vol.35, Providence, R.I.: Am. Math. Soc. 1978Google Scholar
  5. 5.
    Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math.29, 181–202 (1975)Google Scholar
  6. 6.
    Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat.6, 103–144 (1965)Google Scholar
  7. 7.
    Coven, E. M., Reddy, W. L.: Positively expansive maps of compact manifolds. In: Global theory of dynamical systems. Lect. Notes in Mathematics, vol.819, pp. 96–110. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  8. 8.
    Fried, D.: The zeta functions of Ruelle and Selberg I. Ann. Sci. E.N.S.19, 491–517 (1986)Google Scholar
  9. 9.
    Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. vol.16. Providence, R.I., 1955Google Scholar
  10. 10.
    Haydn, N.: Meromorphic extension of the zeta function for Axiom A flows. PreprintGoogle Scholar
  11. 11.
    Manning, A.: Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc.3, 215–220 (1971)Google Scholar
  12. 12.
    Nussbaum, R. D.: The radius of the essential spectrum. Duke Math. J.37, 473–478 (1970)Google Scholar
  13. 13.
    Parry, W., Pollicott, M.: An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. Math.118, 573–591 (1983)Google Scholar
  14. 14.
    Pollicott, M.: A complex Ruelle-Perron-Frobenius theorem and two counterexamples. Ergod. Th. Dynam. Syst.4, 135–146 (1984)Google Scholar
  15. 15.
    Pollicott, M.: Meromorphic extensions of generalized zeta functions. Invent. Math.85, 147–164 (1986)Google Scholar
  16. 16.
    Pollicott, M.: The differential zeta function for Axiom A attractors. PreprintGoogle Scholar
  17. 17.
    Ruelle, D.: A measure associated with Axiom A attractors. Am. J. Math.98, 619–654 (1976)Google Scholar
  18. 18.
    Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. Invent. Math.34, 231–242 (1976)Google Scholar
  19. 19.
    Ruelle, D.: Generalized zeta-functions for axiom A basic sets. Bull. Am. Math. Soc.82, 153–156 (1976)Google Scholar
  20. 20.
    Ruelle, D.: Thermodynamic formalism. Encyclopedia of Math. and its Appl., vol.5, Reading, Mass: Addison-Wesley 1978Google Scholar
  21. 21.
    Ruelle, D.: Repellers for real analytic maps. Ergod. Th. Dynam. Syst.2, 99–107 (1982)Google Scholar
  22. 22.
    Ruelle, D.: One-dimensional Gibbs states and Axiom A diffeomorphisms. J. Differ. Geom.25, 117–137 (1987)Google Scholar
  23. 23.
    Ruelle, D.: Elements of differentiable dynamics and bifurcation theory. Boston: Academic Press 1989Google Scholar
  24. 24.
    Sinai, Ya. G.: Markov partitions andC-diffeomorphisms. Funkts. Analiz i ego Pril.2, 64–89 (1968), English translation: Funct. Anal. Appl.2, 61–82 (1968)Google Scholar
  25. 25.
    Sinai, Ya. G.: Construction of Markov partitions. Funkts. Analiz i ego Pril.2, 70–80 (1968). English translation: Funct. Anal. Appl.2, 245–253 (1968)Google Scholar
  26. 26.
    Sinai, Ya. G.: Gibbs measures in ergodic theory. Usp. Mat. Nauk27, 21–64 (1972), English translation. Russ. Math. Surv.27, 21–69 (1972)Google Scholar
  27. 27.
    Tangerman, F.: Meromorphic continuation of Ruelle zeta functions. Boston University thesis, 1986, (unpublished)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • David Ruelle
    • 1
  1. 1.I.H.E.S.Bures-sur-YvetteFrance

Personalised recommendations