Symplectic capacity and short periodic billiard trajectory
Symplectic capacity and short periodic billiard trajectory
复制标题
辛容量和短周期台球轨迹
DOI：
10.1007/s00209-012-0987-y
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发表时间：
2012
影响因子：
0.8
通讯作者：
Kei Irie
中科院分区：
文献类型：
--
作者：
堀誠;田邊優貴子;工藤栄;山室真澄;佐藤 康彦;田邊優貴子・内田雅己;横山 啓太;Yasuhiko Sato;Keita Yokoyama;佐藤 康彦;Keita Yokoyama;Yasuhiko Sato;Kei Irie
证明了边界光滑的有界域Ω有一个周期台球轨迹，其弹跳次数最多为n+1次，长度小于Cnr(Ω)，其中Cn是一个仅依赖于n的正常数，r(Ω)是Ω中球半径的上确界.这一结果改进了C.Viterbo的结果，他断言Ω有一个长度小于的周期性台球轨迹。为了证明这一结果，我们研究了由辛同调定义的Liouville域的辛容。
We prove that a bounded domainΩinwith smooth boundary has a periodic billiard trajectory with at mostn+ 1 bounce times and of length less thanCnr(Ω), whereCnis a positive constant which depends only onn, andr(Ω) is the supremum of radius of balls inΩ. This result improves the result by C. Viterbo, which asserts thatΩhas a periodic billiard trajectory of length less than. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.