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Geodesic flow on the two-sphere, Part I: Positive measure entropy

Published online by Cambridge University Press:  19 September 2008

Victor J. Donnay
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
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Abstract

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A C metric is constructed on S2 whose geodesic flow has positive measure entropy.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Anosov, D. V.. Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Insl. Math. 90 (1960).Google Scholar
[2]Ballman, W., Brin, M. & Burns, K.. On surfaces with no conjugate points. J. Differential Geometry 25 (1987), 249273.CrossRefGoogle Scholar
[3]Ballman, W., Brin, M. & Eberlein, P.. Structure of manifolds of nonpositive curvature I. Ann. of Math. 12 (2) (1985), 171203.Google Scholar
[4]Bangert, V.. Mather sets for twist maps and geodesies on tori. To appear in Dynamics Reported.Google Scholar
[5]Bunimovich, L. A.. On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65 (1979), 295312.CrossRefGoogle Scholar
[6]Burns, K. & Gerber, M.. Real analytic Bernoulli geodesic flows on S 2. To appear in Ergod. Th. & Dynam. Sys. Vol. 9.Google Scholar
[7]DoCarmo, M. P.. Differential Geometry of Curves and Surfaces, Prentice-Hall: New York, 1976.Google Scholar
[8]Donnay, V. J.. Geodesic flow on the two-sphere with positive entropy. PhD thesis, New York University, 1986.Google Scholar
[9]Donnay, V. J.. Geodesic flow on the two-sphere. Part II: Ergodicity. Dynamical Systems Proc., Univ. of Maryland 19861987. Alexander, J. C. (ed.). Springer Lecture Notes 1342 (1988), 112153.Google Scholar
[10]Eberlein, P.. When is a geodesic flow of Anosov type? J. Differential Geometry 8 (1973), 437463.Google Scholar
[11]Hopf, E.. Closed surfaces without conjugate points. Proc. Nat. Acad. Sci. 34 (1948), 4751.Google Scholar
[12]Lazutkin, V. F.. The existence of caustics for billiards in convex domains. Izvestia Acad. of Sci. Ser. Math. 37 (1) (1973), 186216.Google Scholar
[13]Manning, A.. Curvature bounds for the entropy of the geodesic flow on a surface. J. London Math. Soc. 24 (1981), 351357.Google Scholar
[14]Oseledets, V. I.. A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197221.Google Scholar
[15]Osserman, R.. Lecture at geodesic flow workshop. Cal Tech, 01 1985.Google Scholar
[16]Pesin, Ya. B.. Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surveys 32 (1977), 55114.Google Scholar
[17]Wojtkowski, M.. Invariant families of cones and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 145161.CrossRefGoogle Scholar
[18]Wojtkowski, M.. Principles for the design of billiards with nonvanishing Lyapunov exponent. Commun. Math. Phys. 105 (1986), 391414.Google Scholar