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arXiv 1210.7623math.DS
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非游荡表面流的一个拓扑特征

A topological characterization for non-wandering surface flows

Tomoo Yokoyama

发表信息：
Proc. Amer. Math. Soc. 144 (2016), 315-323
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摘要：

设$v$是紧致曲面上具有任意奇点的连续流。然后我们证明了如果$v$是非游荡的，那么$v$拓扑等价于$C^{\infty}$流，使得没有例外轨道和$\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v) = \{ x \in M \mid ω(x) \cup α(x) \subseteq \mathop{\mathrm{Sing}}(v) \}$，其中$\mathrm{P}$是非闭真轨道的并集，$\sqcup$是不相交的并集符号。此外，$v$是非游荡的当且仅当$\overline{\mathrm{LD}\sqcup \mathop{\mathrm{Per}}(v)} \supseteq M - \mathop{\mathrm{Sing}}(v)$，其中$\mathrm{LD}$是局部稠密轨道的并集，$\overline{A}$是子集$A \subseteq M$的闭包。另一方面，$v$是拓扑传递的当且仅当$v$是非游荡的，使得$ \mathop{\mathrm{int}}(\mathop{\mathrm{Per}}(v) \sqcup \mathop{\mathrm{Sing}}(v)) = \emptyset$和$M - (\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v))$是连通的，其中$\mathrm{int} {A}$是子集$A \subseteq M$的内部。此外，我们还利用$\overline{\mathrm{P}} = \overline{\mathrm{LD}} =\mathbb{T}^2$在$\mathbb{T}^2$上构造了一个光滑流。

英文摘要：

Let $v$ be a continuous flow with arbitrary singularities on a compact surface. Then we show that if $v$ is non-wandering then $v$ is topologically equivalent to a $C^{\infty}$ flow such that there are no exceptional orbits and $\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v) = \{ x \in M \mid ω(x) \cup α(x) \subseteq \mathop{\mathrm{Sing}}(v) \}$, where $\mathrm{P}$ is the union of non-closed proper orbits and $\sqcup$ is the disjoint union symbol. Moreover, $v$ is non-wandering if and only if $\overline{\mathrm{LD}\sqcup \mathop{\mathrm{Per}}(v)} \supseteq M - \mathop{\mathrm{Sing}}(v)$, where $\mathrm{LD}$ is the union of locally dense orbits and $\overline{A}$ is the closure of a subset $A \subseteq M$. On the other hand, $v$ is topologically transitive if and only if $v$ is non-wandering such that $ \mathop{\mathrm{int}}(\mathop{\mathrm{Per}}(v) \sqcup \mathop{\mathrm{Sing}}(v)) = \emptyset$ and $M - (\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v))$ is connected, where $\mathrm{int} {A}$ is the interior of a subset $A \subseteq M$. In addition, we construct a smooth flow on $\mathbb{T}^2$ with $\overline{\mathrm{P}} = \overline{\mathrm{LD}} =\mathbb{T}^2$.

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