Fixed points of symplectic periodic flows
Fixed points of symplectic periodic flows
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DOI：
10.1017/s0143385710000295
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发表时间：
2010-03
影响因子：
0.9
通讯作者：
Á. Pelayo;S. Tolman
中科院分区：
文献类型：
--
作者：
Á. Pelayo;S. Tolman
不动点的研究是几何学和动力学中的经典课题。如果圆在紧辛流形M上以哈密顿方式作用，则经典地知道至少存在不动点{1}{2}，{\dim M}+1$；这是从作用的动量映射的Morse理论得出的。本文利用等变上同调中的Atiyah-Bott-Berline-Vergne(ABBV)局部化证明了这一结论对具有非空不动集的辛圆作用也成立，只要Chern类映射是某处内射的--Chern类映射将该点上作用权的和赋给一个不动点。在没有假设的情况下，我们用不动点个数的较小下界来补充这一结果；从动力系统的观点来看，我们的结果暗示在至少8维的紧致流形上不存在恰好有一个或两个平衡点的辛周期流。
Abstract The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least $\frac {1}{2}\,{\dim M}+1$ fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah–Bott–Berline–Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective—the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.