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Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems

Published online by Cambridge University Press:  19 September 2008

Hans Henrik Rugh
Hans Henrik Rugh
Affiliation:
I.H.E.S., 35 route de Chartres, F-91440 Bures-sur-Yvette, France
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Abstract

We consider a generalized Fredholm determinant d(z) and a generalized Selberg zeta function ζ(ω)−1 for Axiom A diffeomorphisms of a surface and Axiom A flows on three-dimensional manifolds, respectively. We show that d(z) and ζ(ω)−1 extend to entire functions in the complex plane. That the functions are entire and not only meromorphic is proved by a new method, identifying removable singularities by a change of Markov partitions.

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996, pp. 805 - 819
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[1]Bowen, R.. Amer. J. Math. 92 (1970), 725–747.Google Scholar
[2]Bowen, R.. Amer. J. Math. 95 (1973), 429–460.CrossRefGoogle Scholar
[3]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov-diffeomorphisms (Lecture Notes in Mathematics 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[4]Cvitanovié, P. and Eckhardt, B.. Phys. Rev. Lett. 63 (1989), 823.CrossRefGoogle Scholar
[5]Douady, A. and Bonavero, L.. Personal communication. The presented version of the proof is due to A. Douady.Google Scholar
[6]Fried, D.. Ann. scient. Ec. Norm. Sup. 19 (1986), 491.Google Scholar
[7]Fried, D.. The Selberg trace formula and related topics (proc. 1984 conf. at Bowdoin College). Contemp. Math. 53 (1986), 141–163.CrossRefGoogle Scholar
[8]Fried, D.. Meromorphic Zeta Functions for Analytic Flows. Preprint, Boston University, 1993.Google Scholar
[9]Gaspard, P. and Rice, S.. J. Chem. Phys. 90 (1989), 2225, 2242, 2255.CrossRefGoogle Scholar
[10]Ghys, É.. Ann. Sci. École Norm. Sup. (4) 20 (1987), 251–270.Google Scholar
[11]Griffiths, P. and Harris, J.. Principles of Algebraic Geometry. Wiley, New York, 1978.Google Scholar
[12]Gunning, R. C. and Rossi, H.. Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs, NJ, 1965.Google Scholar
[13]Hirsch, M. W. and Pugh, C. C.. Proc. Symp. pure Math., Amer. Math. Soc. 14 (1970), 125–131.Google Scholar
[14]Isola, S.. Comm. Math. Phys. 116 (1988), 33.Google Scholar
[15]Manning, A.. Bull. Amer. Math. Soc. 3 (1971), 215–220;Google Scholar
Ann. Math. 118 (1983), 573.CrossRefGoogle Scholar
[16]Parry, W. and Pollicott, M.. Zeta Functions and the Periodic Structure of Hyperbolic Dynamics (Astérisque 187, 188). Société Mathématique de France, Paris, 1990.Google Scholar
[17]Ruelle, D.. Invent. Math. 34 (1976), 231–242.CrossRefGoogle Scholar
[18]Ruelle, D.. Statistical Mechanics, Thermodynamic Formalism. Addison-Wesley, Reading, 1978.Google Scholar
[19]Ruelle, D.. Comm. Math. Phys. 125(2) (1989), 239.Google Scholar
[20]Ruelle, D.. Publ. Math. IHES. 72 (1990), 175–193.CrossRefGoogle Scholar
[21]Rugh, H. H.. Nonlinearity 5 (1992), 1237–1263.Google Scholar
[22]Selberg, A.. J. Indian. Math. Soc. 20 (1956), 47–87.Google Scholar
[23]Smale, S.. Bull. Am. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar