Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T03:22:51.584Z Has data issue: false hasContentIssue false
  • English
  • Français

On the number of periodic reflecting rays in generic domains

Published online by Cambridge University Press:  19 September 2008

Vesselin M. Petkov  and
Luchezar N. Stojanov
Vesselin M. Petkov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, Bulgaria
Luchezar N. Stojanov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, Bulgaria
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that for generic domains Ω ⊂ ℝn with smooth boundary X for every integer s ≥ 2 there is at most a finite number of periodic reflecting rays with just s reflections on X.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 1 , March 1988 , pp. 81 - 91
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Golubitsky, M. & Guillemin, V.. Stable Mappings and their Singularities. Springer: New York, 1973.CrossRefGoogle Scholar
[2]Katok, A.. Entropy and closed geodesies. Ergod. Th. & Dynam. Sys. 2 (1982), 339–367.CrossRefGoogle Scholar
[3]Katok, A. & Strelcyn, J.-M.. The estimation of entropy from above for differentiable maps with singularities. (Preprint, 1981)Google Scholar
[4]Katok, A. & Strelcyn, J.-M.. Smooth maps with singularities: invariant manifolds, entropy and billiards. Lecture Notes in Mathematics, Volume 1222, Springer: Berlin, 1986.Google Scholar
[5]Margulis, G. A.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Funct. Anal. Appl. 3 (1969), 335–336.Google Scholar
[6]Melrose, R. & Sjöstrand, J.. Singularities in boundary value problems, I, II. Comm. Pure Appl. Math. 31 (1978), 593–617 and 35 (1982), 129–168.Google Scholar
[7]Petkov, V.. Propriétes génériques des rayons réflechissants et applications aux problèmes spectraux. Séminaire Bony-Sjöstrand-Meyer. Ecole Polytechnique, Centre de Math., Exposé XII (1984–1985).Google Scholar
[8]Petkov, V.. High frequency asymptotics of the scattering amplitude for non-convex bodies. Comm. Partial Diff. Equations. 5 (1980), 293–329.Google Scholar
[9]Petkov, V. & Stojanov, L.. Periods of multiple reflecting geodesies and inverse spectral results. Amer. J. Math. 109 (1987), 619–668.CrossRefGoogle Scholar
[10]Petkov, V. & Stojanov, L.. Spectrum of the Poincaré map for periodic reflecting rays in generic domains. Math. Z. 194 (1987), 505–518.Google Scholar
[11]Petkov, V. & Stojanov, L.. Propriétés génériques de l' application de Poincaré et des geodesiques périodiques généralisées. Seminaire Equations aux Dérivées Partielles, Ecole Polytechnique, Centre de Math., Exposé XI (1985–1986).Google Scholar
[12]Petkov, V. & Stojanov, L.. Periodic geodesies of generic non-convex domains in ℝ2 and the Poisson relation. Bull. Amer. Math. Soc. 15 (1986), No. 1, 88–90.Google Scholar
[13]Stojanov, L.. Generic properties of periodic reflecting rays. Ergod. Th. Dynam. Sys. 7 (1987).CrossRefGoogle Scholar
[14]Stojanov, L.. Generic properites of scattered rays. C.R. Acad. Bulgare Sci. 39 (1986), 13–14.Google Scholar