Dynamical Zeta Functions: Where Do They Come from and What Are They Good for ?

  • David Ruelle
Conference paper


The properties and usefulness of dynamical zeta functions associated with maps and flows are discussed, and they are compared with the more traditional number-theoretic zeta functions.


Zeta Function Periodic Point Closed Geodesic Riemann Zeta Function Geodesic Flow 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artin, M., Mazur, B.: On periodic points. Ann.of Math. (2) 81, pp. 82–99 (1965)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Artuso, R., Aurell, E., Cvitanovic, P.: Recycling of strange sets: I. Cycle expansions, II. Applications. Nonlinearity 3, pp. 325–359, 361–386 (1990)MathSciNetCrossRefMATHADSGoogle Scholar
  3. 3.
    Baladi, V. and Keller, G.: Zeta functions and transfer operators for piece-wise monotone transformations. Comm.Math.Phys. 127, pp. 459–477 (1990)MathSciNetCrossRefMATHADSGoogle Scholar
  4. 4.
    Cartier, P. and Voros, A.: Une nouvelle interprétation de la formule des traces de Selberg. C.R. Acad.Sci.Paris, 307, Ser. I, pp. 143–148 (1988)MathSciNetMATHGoogle Scholar
  5. 5.
    Deligne, P.: La conjecture de Weyl. Publ.math. IHES, 43, pp. 273–307 (1974)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fried, D.: The zeta functions of Ruelle and Selberg I. Ann.Sci. E.N.S. 19, pp. 491–517 (1986)MathSciNetMATHGoogle Scholar
  7. 7.
    Grothendick, A.: La théorie de Fredholm. Bull.Soc.Math.France 84, pp. 319–384 (1956)MathSciNetGoogle Scholar
  8. 8.
    Guckenheimer, J.: Axiom A + no cycles ⇒ ζ f (t) rational. Bull.AMS 76, pp. 592–594 (1970)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Haydn, N.: Meromorphic extension of the zeta function for Axiom A flows. Ergodic Therory Dynamical Systems 10, pp. 347–360 (1990)MathSciNetMATHGoogle Scholar
  10. 10.
    Korevaar, J.: On Newman’s quick way to the prime number theorem. Math. Intelligencer 4, pp.108–115 (1982)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Manning, A.: Axiom A diffeomorphisms have rational zeta functions. Bull. London Math.Soc. 3, pp. 215–220 (1971)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Mayer, D.: On a zeta function related to the continued fraction transformation. Bull.Soc.Math. France 104, pp. 195–203 (1976)MATHADSGoogle Scholar
  13. 13.
    Mayer, D.: On the thermodynamic formalism for the Gauss map. Comm. Math.Phys. 130, pp. 311–333 (1990)MathSciNetCrossRefMATHADSGoogle Scholar
  14. 14.
    Mayer, D.: The thermodynamic formalism approach to Selberg’s zeta function for PSL (2, ℤ). Bull.Amer.Math.Soc. 25, pp. 55–60 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nihon Sugakkai, ed.: Encyclopedic dictionary of mathematics. MIT Press, Cambridge, Mass., (1977)Google Scholar
  16. 16.
    Parry, W., Pollicott, M.: An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann.Math. 118, pp. 573–591 (1983)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188, pp. 1–268 (1990)Google Scholar
  18. 18.
    Pollicott, M.: Meromorphic extensions of generalised zeta functions. Invent.Math. 85, pp. 147–164 (1986)MathSciNetCrossRefMATHADSGoogle Scholar
  19. 19.
    Pollicott, M.: Distribution of closed geodesics on the modular surface and quadratic irrationals. Bull.Soc.Math. France 114, pp. 431–446 (1986)MathSciNetMATHGoogle Scholar
  20. 20.
    Pollicott, M.: Margulis distributions for Anosov flows. Comm.Math.Phys. 113, pp. 137–154 (1987)MathSciNetCrossRefMATHADSGoogle Scholar
  21. 21.
    Ruelle, D.: Generalized zeta-functions for axiom A basic sets. Bull.Amer. Math.Soc. 82, pp. 153–156 (1976)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ruelle, D.: Zeta functions for expanding maps and Anosov flows. Invent. Math. 34, pp. 231–242 (1976)MathSciNetCrossRefMATHADSGoogle Scholar
  23. 23.
    Ruelle, D.: Zeta functions and statistical mechanics. Astérisque 40, pp. 167–176 (1976)MathSciNetGoogle Scholar
  24. 24.
    Ruelle, D.: Thermodynamic Formalism. Encyclopedia of Math, and its Appl. Vol. 5, Addison-Wesley, Reading, Mass. (1978)Google Scholar
  25. 25.
    Ruelle, D.: The thermodynamic formalism for expanding maps. Comm. Math.Phys. 125, pp. 239–262 (1989)MathSciNetCrossRefMATHADSGoogle Scholar
  26. 26.
    Ruelle, D.: An extension of the theory of Fredholm determinants. Publ. Math. IHES, 72, pp. 175–193 (1990)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ruelle, D.: Spectral properties of a class of operators associated with con-formal maps in two dimensions (preprint)Google Scholar
  28. 28.
    Sarnak, P.: Class numbers of indefinite binary quadratic forms. J.Number Theory 15, pp. 229–247 (1982)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Sarnak, P.: Determinants of Laplacians. Comm.Math.Phys. 110, pp. 113–120 (1987)MathSciNetCrossRefMATHADSGoogle Scholar
  30. 30.
    Selberg, A.: Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J.Indian Math.Soc. 20, pp. 47–87 (1956)MathSciNetMATHGoogle Scholar
  31. 31.
    Series, C.: The modular surface and continued fractions. J.London Math. Soc. (2) 31, pp. 69–80 (1985)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Smale, S.: Differentiable dynamical systems. Bull.Amer.Math.Soc.73, pp. 747–817 (1967)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tangerman, F.: Meromorphic continuation of Ruelle zeta functions. Boston University thesis (1986) unpublishedGoogle Scholar
  34. 34.
    Weil, A.: Numbers of solutions of equations in finite fields. Bull.Amer.Math. Soc. 55, pp. 497–508 (1949)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • David Ruelle
    • 1
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

Personalised recommendations