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Glancing billiards

Published online by Cambridge University Press:  19 September 2008

John N. Mather
John N. Mather
Affiliation:
Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
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Abstract

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Consider the billiard ball problem in an open, convex, bounded region of the plane whose boundary is C2 and has at least one point of zero curvature. Then there are trajectories which come arbitrarily close to being positively tangent to the boundary and also come arbitrarily close to being negatively tangent to the boundary.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 3-4 , December 1982 , pp. 397 - 403
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Birkhoff, G. D.. Surface transformations and their dynamical applications. Ada Math. 43 (1922), 1–119.Google Scholar
Reprinted in Collected Mathematical Papers, vol. II. Amer. Math. Soc: New York, 1950, pp. 111–229.Google Scholar
[2]Birkhoff, G. D.. Sur quelques courbes fermeés remarquables. Bull. Soc. Math, de France 60 (1932), 1–26.Google Scholar
Reprinted in Collected Mathematical Papers, vol. II. Amer. Math. Soc: New York, 1950, pp. 418–443.Google Scholar
[3]Bunimovich, L. A.. On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65 (1979), 295–312.CrossRefGoogle Scholar
[4]Lazutkin, V. F.. The existence of caustics for a billiard problem in a convex domain. Math. USSR Izvestija 7 (1973), 185–214. Translation of Izvestija, Mathematical Series, Academy of Sciences of the USSR, 37, 1973.CrossRefGoogle Scholar
[5]Mather, J. N.. Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. (To appear in Topology.)Google Scholar
[6]Newman, R. P. A. C. & Percival, I. C.. Bounds on domains in phase space for Lagrangian systems with discrete time. Preprint, Queen Mary College.Google Scholar
[7]Sinai, Ya. G.. Introduction to Ergodic Theory, Math. Notes, Princeton University Press: Princeton 1977. (Translated from Russian by V. Scheffer.)Google Scholar