Hostname: page-component-65f69f4695-xtrld Total loading time: 0 Render date: 2025-07-01T06:52:19.285Z Has data issue: false hasContentIssue false
  • English
  • Français

Meromorphic extension of the zeta function for Axiom A flows

Published online by Cambridge University Press:  19 September 2008

Nicolai T. A. Haydn
Nicolai T. A. Haydn
Affiliation:
Department of Mathematics, University of Toronto, Toronto M5S 1A1, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the meromorphicity of the zeta function on shifts of finite type for Hölder continuous functions assuming that the essential spectrum of the associated Ruelle operator is contained in the open unit disc. This result allows to extend the region of meromorphicity of the zeta function for Axiom A flows by a strip whose width is determined by the contraction rate of the flow.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 347 - 360
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Abramov, L. M.. On the entropy of a flow. Dokl. Akad. Nauk. SSSR 128 (1959), 873–876.Google Scholar
Amer. Math. Soc. Transl. 2 49 (1966), 167–170.Google Scholar
[2]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. S.L.N. No. 470. Springer, New York, 1975.CrossRefGoogle Scholar
[3]Bowen, R.. Sybolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–459.Google Scholar
[4]Gallovotti, G.. Funzione zeta ed insiemi basilar; Accad. Linei. Rend. Sc. fisimat. e nat. 61 (1976), 309–317.Google Scholar
[5]Hofbauer, F. & Keller, G.. Zeta functions and transfer operators for piecewise linear transformations; J. reine angew. Math. 352 (1984), 100–113.Google Scholar
[6]Manning, A.. Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc. 3 (1971), 215–220.CrossRefGoogle Scholar
[7]Nussbaum, R. D.. The radius of the essential spectrum. Duke Math. J. 37 (1970), 473–478.CrossRefGoogle Scholar
[8]Parry, W.. An analogue of the prime number theorem for shifts of finite type and their suspension. Israel J. Math. 45 (1983), 41–52.CrossRefGoogle Scholar
[9]Parry, W.. Bowen's equidistribution theory and the Dirichlet density theorem. Ergod. Th. & Dynam. Syst. 4 (1984), 117–134.Google Scholar
[10]Pollicott, M.. Meromorphic extensions of generalised zeta functions. Invent. Math. 85 (1986), 147–164.CrossRefGoogle Scholar
[11]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading, 1978.Google Scholar
[12]Ruelle, D.. One dimensional Gibbs' states and Axiom A diffeomorphisms. J. Diff. Geom. 25 (1987), 117–137.Google Scholar
[13]Sinai, Ya. G.. Gibbs' measures in ergodic theory. Russ. Math. Surv. 27(4) (1972), 21–69.CrossRefGoogle Scholar
[14]Walters, P.. An Introduction to Ergodic Theory. G.T.M. No. 79. Springer, New York, 1981.Google Scholar