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Entropy estimates for geodesic flows

Published online by Cambridge University Press:  19 September 2008

P. Sarnak
P. Sarnak
Affiliation:
Courant Institute for Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA
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Abstract

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Let M be a compact Riemannian manifold of (variable) negative curvature. Let h be the topological entropy and hμ the measure entropy for the geodesic flow on the unit tangent bundle to M. Estimates for h and hμ in terms of the ‘geometry’ of M are derived. Connections with and applications to other geometric questions are discussed.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 3-4 , December 1982 , pp. 513 - 524
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Berger, M.. La Spectre d'une variete Riemmanienne. Springer Lecture Notes no. 194.Google Scholar
[2]Bowen, R.. On axiom A diffeomorphisms. Conference Board of Math. Sciences. Regional Conf. Ser. in Math., Amer. Math. Soc: Rhode Island, 35 (1978), 17–20.Google Scholar
[3]Eberlein, P.. J. Diff. Geometry 8 (1973), 437–463.Google Scholar
[4]Guillemin, V. & Duistermaat, J.. The spectrum of a positive elliptic operator and periodic bicharacteristics. Invent. Math. 29 (1975), 37–79.Google Scholar
[5]Goodwyn, W.. Comparing topological entropy with measure entropy. Amer. J. of Math. 94 (1972), 336.CrossRefGoogle Scholar
[6]Katok, A.. Entropy and closed geodesies. Ergod. Th. & Dynam. Sys. 2 (1982), 339.CrossRefGoogle Scholar
[7]Manning, A.. Topological entropy for geodesic flows, Annal Math. 110 (1979), 567–573.CrossRefGoogle Scholar
[8]Mckean, H.. An upper bound for the spectrum of A on a manifold of negative curvature, J. Diff. Geom. 4 (1970), 359.Google Scholar
[9]Mckean, H. & Singer, I.. Curvature and eigenvalues of the Laplacian. J. Diff. Geom. 1 (1967), 43–67.Google Scholar
[10]Margulis, G.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Func. Anal, and Appl. 3 (1969), 335–336.Google Scholar
[11]Pesin, Ya.. Russian Math. Surveys 32–4 (1977), 55–114.Google Scholar
[12]Reed, M. & Simon, B.. Meth. Math. Phys. vol. 4, Academic Press: New York, 1978, p. 279.Google Scholar
[13]Sinai, Ja.. Introduction to Ergodic Theory. Princeton Univ. Press: Princeton, 1976. p. 119.Google Scholar
[14]Sinai, Ja.. The asymptotic behaviour of the number of closed geodesies. A.M.S. Transl. 73 (1968), 227–250.Google Scholar