平面向量场退化中心奇点的对称性
Symmetries of degenerate center singularities of plane vector fields
摘要：
设$D$是以原点$O$为中心的平面上的一个闭单位$2$ -圆盘，$F$是一个光滑向量场，使得$O$是$F$的唯一奇点，并且$F$的所有其他轨道都是围绕$O$缠绕一次的简单闭曲线。因此，拓扑上$O$是一个“中心”奇点。还令$\mathrm{Diff}(F)$是保持$F$的方向和轨道的$D$的所有同构的群。在arXiv：0907.0359中，作者在假设$F$在$O$的$1$ -喷流是非退化的情况下描述了$\mathrm{Diff}(F)$的同伦类型。本文考虑退化情形。在额外的“非退化假设”下，对$F$的路径分量$\mathrm{Diff}(F)$相对于不同的弱拓扑进行了描述。
英文摘要：
Let $D$ be a closed unit $2$-disk on the plane centered at the origin $O$, and $F$ be a smooth vector field 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 also $\mathrm{Diff}(F)$ be the group of all diffeomorphisms of $D$ which preserve orientation and orbits of $F$. In arXiv:0907.0359 the author described the homotopy type of $\mathrm{Diff}(F)$ under assumption that the $1$-jet of $F$ at $O$ is non-degenerate. In this paper degenerate case is considered. Under additional "non-degeneracy assumptions" on $F$ the path components of $\mathrm{Diff}(F)$ with respect to distinct weak topologies are described.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
