表面流动正、负长时间行为的依赖性
Dependency of the positive and negative long-time behaviors of flows on surfaces
摘要：
长时间行为是动力系统最基本的性质之一。庞加莱-本迪克松定理利用$ω$ -极限集描述了曲面上流动的极限行为。本文论证了正长时行为和负长时行为不是独立的。实际上，我们证明了曲面上流点的$ω$ -极限集与$α$ -极限集之间的依赖关系，这部分地推广了庞加莱-本迪克松定理.应用相关结果解决了紧致曲面上的保面积流（或更一般的非游荡流）中出现何种$ω$ -极限集的问题，证明了紧致曲面上具有任意多个奇点的保面积流的任何非闭轨道的$ω$ -极限集要么是奇点的子集，要么是局部稠密Q-集.此外，我们显示的野生手术，以增加全不连通的奇点和驯服的手术，以增加许多奇点的表面上的流动。
英文摘要：
Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors of flows on surfaces are captured by the Poincaré-Bendixson theorem using the $ω$-limit sets. This paper demonstrates that the positive and negative long-time behaviors are not independent. In fact, we show the dependence between the $ω$-limit sets and the $α$-limit sets of points of flows on surfaces, which partially generalizes the Poincaré-Bendixson theorem. Applying the dependency result to solve what kinds of the $ω$-limit sets appear in the area-preserving (or, more generally, non-wandering) flows on compact surfaces, we show that the $ω$-limit set of any non-closed orbit of such a flow with arbitrarily many singular points on a compact surface is either a subset of singular points or a locally dense Q-set. Moreover, we show the wildness of surgeries to add totally disconnected singular points and the tameness of those to add finitely many singular points for flows on surfaces.
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