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Approximate solutions of cohomological equations associated with some Anosov flows

Published online by Cambridge University Press:  19 September 2008

Svetlana Katok
Svetlana Katok
Affiliation:
Department of Mathematics, University of California, Santa Cruz, Santa Cruz, CA 95064, USA
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Abstract

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The Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 367 - 379
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. of Math. 90 (1967).Google Scholar
[2]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.CrossRefGoogle Scholar
[3]Guillemin, V. & Kazhdan, D.. On the cohomology of certain dynamical systems. Topology 19 (1980), 291–299.CrossRefGoogle Scholar
[4]Hopf, E.. Statistik der geodätischer Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.Google Scholar
[5]Hopf, E.. Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II. Math. Ann. 117 No. 4 (1940), 590–608.CrossRefGoogle Scholar
[6]Hurder, S. & Katok, A.. Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, to appear, Publ. Math. IHES.Google Scholar
[7]Livčic, A. N.. Some homology properties of U-systems. Mat. Zametki 10 (1971), 555–564;Google Scholar
Math. Notes 10 (1971), 758–763.CrossRefGoogle Scholar
[8]de la Llave, R., Marco, J. & Moriyon, R.. Canonical perturbation theory of Anosov systems and regularity results for Livčic cohomology equation. Ann. of Math. 123 No. 3 (1986), 537–612.CrossRefGoogle Scholar