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On Moser’s boundedness problem of dual Billiards

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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.

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[1] J. Moser . Is the Solar System stable? Math. Intelligencer 1 (1978), 6571.

[2] F. Vivaldi and A. V. Shaidenko . Global stability of a class of discontinous dual billiards. Comm. Math. Phys. 110 (1987), 625640.

[4] E. Gutkin and N. Simanyi . Dual polygonal billiards and necklace dynamics. Comm. Math. Phys. 143 (1992), 431449.

[5] S. Tabachnikov . On the dual billiard problem. Adv. Math. 115 (1995), 221249.

[10] R. Schwartz . Unbounded orbits for outer billiards. I. J. Mod. Dyn. 1 (2007), 371424.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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