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Generic properties of periodic reflecting rays

Published online by Cambridge University Press:  19 September 2008

Luchezar Stojanov
Luchezar Stojanov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, Bulgaria
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Abstract

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It is shown that for generic domains D in n, n ≥ 2, every periodic billiard trajectory in D passes only once through each of its reflection points, and any two different periodic billiard trajectories in D have no common reflection point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 4 , December 1987 , pp. 597 - 609
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Golubitsky, M. & Guillemin, V.. Stable Mappings and their Singularities. Springer: New York, 1973.CrossRefGoogle Scholar
[2]Guillemin, V. & Melrose, R.. The Poisson summation formula for manifolds with boundary. Adv. in Math. 38 (1979), 204–233.CrossRefGoogle Scholar
[3]Hirsch, M.. Differential Topology. Springer: New York, 1976.CrossRefGoogle Scholar
[4]Kac, M.. Can one hear the shape of a drum? Amer. Math. Soc. Monthly 73 (1966), 1–23.CrossRefGoogle Scholar
[5]Lazutkin, V.. Convex Billiards and Eigenfunctions of the Laplace Operator. Ed. Leningrad University, 1981 (in Russian).Google Scholar
[6]Marvizi, S. & Melrose, R.. Spectral invariants of convex planar regions. J. Diff. Geometry 17 (1982), 475–502.Google Scholar
[7]Petkov, V.. Poisson relation for manifolds with boundary. Symposium and Workshop on Hyperbolic Equations and Related Topics. Kyoto, 1984, 317–327.Google Scholar
[8]Petkov, V.. Propriétés génériques des rayons refleckissants et applications aux problèmes spectraux. Seminaire Bony-Sjöstrand-Meyer, Ecole Polytechnique, Centre de Mathématiques, Exposè XII (1984–1985).Google Scholar
[9]Petkov, V. & Stojanov, L.. Periods of multiple reflecting geodesies and inverse spectral results. Amer. J. Math. 109 (1987).CrossRefGoogle Scholar
[10]Petkov, V. & Stojanov, L.. Periodic geodesies of generic non-convex domains in çoes de Matematica, Universidade Federal de Pernambuco: Recife-Brasil, No. 138 (1985).Google Scholar
[11]Petkov, V. & Stojanov, L.. Spectrum of the Poincaré map for periodic reflecting rays in generic domains. Math. Z. 194 (1987), 505–517.CrossRefGoogle Scholar