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arXiv · 1010.3170

Symplectic capacity and short periodic billiard trajectory

Kei Irie
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Abstract

We prove that a bounded domain $Ω$ in $\R^n$ 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 $C'_n \vol(Ω)^{1/n}$. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.

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Kei Irie. 2012-02-06. Symplectic capacity and short periodic billiard trajectory. https://doi.org/10.1007/s00209-012-0987-y

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