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On Moser’s boundedness problem of dual Billiards

Published online by Cambridge University Press:  01 April 2009

LONG LI
LONG LI*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China (email: [email protected])
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Abstract

In this paper, the boundedness of the dual billiard orbits of trapezoids is obtained. This gives an example showing that there are also polygons, which are neither rational nor quasi-rational, with all of their dual billiard orbits bounded. Moreover, most of the orbits are Poisson-stable, which means that most points will come sufficiently near to their original positions under iterations of the dual billiard map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 613 - 635
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Moser, J.. Is the Solar System stable? Math. Intelligencer 1 (1978), 65–71.CrossRefGoogle Scholar
[2]Vivaldi, F. and Shaidenko, A. V.. Global stability of a class of discontinous dual billiards. Comm. Math. Phys. 110 (1987), 625–640.CrossRefGoogle Scholar
[3]Kolodziej, R.. The antibilliard outside a polygon. Bull. Pol. Acad. Sci. 37 (1989), 163–168.Google Scholar
[4]Gutkin, E. and Simanyi, N.. Dual polygonal billiards and necklace dynamics. Comm. Math. Phys. 143 (1992), 431–449.CrossRefGoogle Scholar
[5]Tabachnikov, S.. On the dual billiard problem. Adv. Math. 115 (1995), 221–249.CrossRefGoogle Scholar
[6]Tabachnikov, S.. Billiards, S M F. Panor. Synthèses 1 (1995).Google Scholar
[7]Tabachnikov, S.. Geometry and Billiards. American Mathematical Society, Providence, RI, 2005.CrossRefGoogle Scholar
[8]Dogru, F. and Tabachnikov, S.. Dual billiards. Math. Intelligencer 27(4) (2005), 18–25.Google Scholar
[9]Genin, D.. Regular and chaotic dynamics of outer billiards. PhD Thesis, Penn State University, 2005.Google Scholar
[10]Schwartz, R.. Unbounded orbits for outer billiards. I. J. Mod. Dyn. 1 (2007), 371–424.CrossRefGoogle Scholar