Decompositions of surface flows
Decompositions of surface flows
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表面流的分解
DOI：
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发表时间：
2017
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通讯作者：
T. Yokoyama
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作者：
T. Yokoyama
我们构造了有限类型面流的一个完全有限不变量。特别地，该不变量对于Morse-Smer流和“一般”哈密顿流是完全的。事实上，虽然奇点的退化意味着无数的局部拓扑等价类和极小流拓扑等价类的集合。Denjoy流)是不可数的，我们用有限标号图来计数紧曲面上至多有有限个极限环但没有非闭循环轨道或退化奇异点的流拓扑等价类的集合。为了计数这类流，我们描述了紧致曲面上没有退化奇点的流的边界点的性质。特别地，我们证明了“ss-多鞍点连通图”的补集的每个连通分支要么是开盘、开环、环面、Klein瓶、开M带，要么是由局部稠密轨道组成的开本质子集。此外，我们推广了紧致曲面上具有任意多个奇点的流的Poincar-e-Bendixson定理。事实上，任何非闭轨道的$-极限集要么是奇点的无处稠密子集，要么是极限环，要么是极限“拟圈”，要么是局部稠密Q-集，要么是“准Q集”。此外，对于这种表面流，我们刻画了对应于非游荡集的闭轨并闭合的充要条件。
We construct a complete finite invariant for surface flows of finite type. In particular, the invariant is complete for Morse-Smale flows and "generic" Hamiltonian flows. In fact, although degeneracy of singular points implies uncountably many local topological equivalence classes and the set of topological equivalence classes of minimal flows (resp. Denjoy flows) on a torus is uncountable, we enumerate the set of topological equivalence classes of flows with at most finitely many limit cycles but without non-closed recurrent orbits or degenerate singular points on a compact surface using finite labelled graphs. To enumerate such flows, we describe properties of border points of a flow without degenerate singular points on a compact surface. In particular, we show that each connected component of the complement of the "ss-multi-saddle connection diagram" is either an open disk, an open annulus, a torus, a Klein bottle, an open M\"obius band, or an open essential subset consisting of locally dense orbits. Moreover, we generalize the Poincar\'e-Bendixson theorem for a flow with arbitrarily many singular points on a compact surface. In fact, the $\omega$-limit set of any non-closed orbit is either a nowhere dense subset of singular points, a limit cycle, a limit "quasi-circuit", a locally dense Q-set, or a "quasi-Q set". In addition, for such a surface flow, we characterize the necessary and sufficient conditions for the closure of the union of closed orbits corresponding to the non-wandering set.