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 C n r(Ω), where C n 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albers, P., Mazzucchelli, M.: Periodic bounce orbits of prescribed energy. Int. Math. Res. Notices. doi:10.1093/imrn/rnq193 (2010)
Benci V., Giannoni F.: Periodic bounce trajectories with a low number of bounce points. Ann. Inst. Henri Poincaré Anal. Non Linéaire 6(1), 73–93 (1989)
Cieliebak K.: Handle attaching in symplectic homology and the Chord Conjecture. J. Eur. Math. Soc. 4, 115–142 (2002)
Cieliebak K., Floer A., Hofer H., Wysocki K.: Applications of symplectic homology II: stability of the action spectrum. Math. Z. 223, 27–45 (1996)
Long Y.: Index Theory for Symplectic Paths with Applications. Progr. Math., vol. 207. Birkhäuser, Basel (2002)
Oancea A.: A survery of Floer homology for manifolds with contact type boundary or symplectic homology. Ensaios Math. 7, 51–91 (2004)
Viterbo C.: Functors and computations in Floer homology I. Geom. Funct. Anal. 9, 985–1033 (1999)
Viterbo C.: Metric and isoperimetic problems in symplectic geometry. J. Am. Math. Soc. 13(2), 411–431 (2000)
Weinstein A.: Contact surgery and symplectic handlebodies. Hokk. Math. J. 20, 241–251 (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Irie, K. Symplectic capacity and short periodic billiard trajectory. Math. Z. 272, 1291–1320 (2012). https://doi.org/10.1007/s00209-012-0987-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-012-0987-y