Mathematische Zeitschrift

, Volume 272, Issue 3–4, pp 1291–1320 | Cite as

Symplectic capacity and short periodic billiard trajectory

Article

Abstract

We prove that a bounded domain Ω in \({\mathbb R^n}\) with smooth boundary has a periodic billiard trajectory with at most n + 1 bounce times and of length less than Cnr(Ω), where Cn is a positive constant which depends only on n, and r(Ω) is the supremum of radius of balls in Ω. This result improves the result by C. Viterbo, which asserts that Ω has a periodic billiard trajectory of length less than \({C'_{n} {\rm vol}(\Omega)^{1/n}}\). To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.

Keywords

Symplectic capacity Periodic billiard trajectory Symplectic homology 

Mathematics Subject Classification (2010)

Primary 34C25 Secondary 53D40 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

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