Mathematische Zeitschrift

, Volume 272, Issue 3, pp 1291–1320

Symplectic capacity and short periodic billiard trajectory

Authors

    • Department of Mathematics, Faculty of ScienceKyoto University
Article

DOI: 10.1007/s00209-012-0987-y

Cite this article as:
Irie, K. Math. Z. (2012) 272: 1291. doi:10.1007/s00209-012-0987-y

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 capacityPeriodic billiard trajectorySymplectic homology

Mathematics Subject Classification (2010)

Primary 34C25Secondary 53D40

Copyright information

© Springer-Verlag 2012