辛容量与短周期台球轨迹
Symplectic capacity and short periodic billiard trajectory
摘要：
本文证明了$\R^n$中具有光滑边界的有界区域$Ω$有一个周期台球轨迹，它至多有$n+1$反弹次数，且长度小于$C_n r(Ω)$，其中$C_n$是一个只依赖于$n$的正常数，$r(Ω)$是$Ω$中球半径的上确界.这个结果改进了C.Viterbo的结果，后者断言$Ω$具有长度小于$C'_n \vol(Ω)^{1/n}$的周期台球轨迹。为了证明这个结果，我们研究了Liouville域的辛容度，它是通过辛同调定义的。
英文摘要：
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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