Advertisement

An extension of the theory of Fredholm determinants

  • David Ruelle
Article

Abstract

Analytic functions are introduced, which are analogous to the Fredholm determinant, but may have only finite radius of convergence. These functions are associated with operators of the form ε μ(dω) ℒω, where ℒω φ(x) = ϕω(x). φ(ψω x), , φ belongs to a space of Hölder or C r functions, ϕω is Hölder or C r , and ψω is a contraction or a C r contraction. The results obtained extend earlier results by Haydn, Pollicott, Tangerman and the author on zeta functions of expanding maps.

Keywords

Vector Bundle Taylor Expansion Zeta Function Spectral Radius Fredholm Determinant 
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]
    V. Baladi andG. Keller, Zeta functions and transfer operators for piecewise monotone transformations,Commun. Math. Phys.,127 (1990), 459–477.zbMATHCrossRefGoogle Scholar
  2. [2]
    R. Bowen, Markov positions for Axiom A diffeomorphisms,Trans. Amer. Math. Soc.,154 (1971), 377–397.zbMATHCrossRefGoogle Scholar
  3. [3]
    D. Fried, The zeta functions of Ruelle and Selberg, I.,Ann. Sci. E.N.S.,19 (1986), 491–517.zbMATHGoogle Scholar
  4. [4]
    A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires,Memoirs of the Amer. Math. Soc.,16, Providence, RI, 1955.Google Scholar
  5. [5]
    A. Grothendieck, La théorie de Fredholm,Bull. Soc. Math. France,84 (1956), 319–384.zbMATHGoogle Scholar
  6. [6]
    N. Haydn, Meromorphic extension of the zeta function for Axiom A flows,Ergod. Th. and Dynam. Syst.,10 (1990), 347–360.zbMATHGoogle Scholar
  7. [7]
    D. Mayer, On a ζ function related to the continued fraction transformation,Bull. Soc. Math. France,104 (1976), 195–203.zbMATHGoogle Scholar
  8. [8]
    R. D. Nussbaum, The radius of the essential spectrum,Duke Math. J.,37 (1970), 473–478.zbMATHCrossRefGoogle Scholar
  9. [9]
    M. Pollicott, A complex Ruelle-Perron-Frobenius theorem and two counterexamples,Ergod. Th. and Dynam. Syst.,4 (1984), 135–146.zbMATHGoogle Scholar
  10. [10]
    F. Riesz etB. Sz-Nagy,Leçons d’analyse fonctionnelle, 3e éd., Académie des Sciences de Hongrie, 1955.Google Scholar
  11. [11]
    D. Ruelle, Zeta-functions for expanding maps and Anosov flows,Inventiones Math.,34 (1976), 231–242.zbMATHCrossRefGoogle Scholar
  12. [12]
    D. Ruelle, Thermodynamic Formalism,Encyclopedia of Math and its Appl.,5, Addison-Wesley, Reading, Massachusetts, 1978.Google Scholar
  13. [13]
    D. Ruelle, The thermodynamic formalism for expanding maps,Commun. Math. Phys.,125 (1989), 239–262.zbMATHCrossRefGoogle Scholar
  14. [14]
    Ia. G. Sinai, Construction of Markov partitions,Funkts. Analiz i ego Pril.,2, No. 3 (1968), 70–80. English translation:Functional Anal. Appl.,2 (1968), 245–253.Google Scholar
  15. [15]
    F. Tangerman,Meromorphic continuation of Ruelle zeta functions, Boston University Thesis, 1986 (unpublished).Google Scholar

Copyright information

© Publications Mathématiques de L’I.É.E.S. 1990

Authors and Affiliations

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

Personalised recommendations