Symplectic capacity and short periodic billiard trajectory

Abstract

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

32 pages, final version with minor modifications. Published online in Mathematische Zeitschrift

References in corpus (2)

• Functors and Computations in Floer homology with Applications Part II
• Periodic bounce orbits of prescribed energy

Cited by in corpus (6)

• Periodic ellipsoidal billiard trajectories and extremal polynomials
• When symplectic topology meets Banach space geometry
• Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications
• Symplectic homology of disc cotangent bundles of domains in Euclidean space
• Displacement energy of unit disk cotangent bundles
• A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories