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Mathematics  2010 

Symplectic capacity and short periodic billiard trajectory

DOI: 10.1007/s00209-012-0987-y

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Abstract:

We prove that a bounded domain $\Omega$ in $\R^n$ with smooth boundary has a periodic billiard trajectory with at most $n+1$ bounce times and of length less than $C_n r(\Omega)$, 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 $C'_n \vol(\Omega)^{1/n}$. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.

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