平面向量场中心奇点的对称性
Symmetries of center singularities of plane vector fields
摘要：
设D是以原点0为中心的平面上的闭单位2-圆盘，F是D上的光滑向量场，使得O是F的唯一奇点，F的所有其它轨道都是绕O一次的简单闭曲线.因此，拓扑O是一个“中心”奇点。设p：D\0-->（0，+ \infty）是将每个$z\not=O$与其相对于F的周期相关联的函数。这个函数在O处可以是不连续的。设Diff（F）是D中所有保持F的方向和轨道的自同态的群。我们证明了p光滑扩张到D的全部当且仅当F在原点的1-jet是非退化线性映射，并且在这种情况下Diff（F）同伦等价于圆。
英文摘要：
Let D be a closed unit 2-disk on the plane centered at the origin 0, and F be a smooth vector field on D such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a ``center'' singularity. Let p:D\0-->(0,+\infty) be the function associating to each $z\not=O$ its period with respect to F. This function can be discontinuous at O. Let Diff(F) be the group of all diffeomorphisms of D which preserve orientation and orbits of F. We prove that p smoothly extends to all of D if and only if the 1-jet of F at the origin is a non-degenerate linear map, and that in this case Diff(F) is homotopy equivalent to the circle.
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