A topological characterization for non-wandering surface flows

Abstract:

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 $\mathrm {Per}(v)$ is open, there are no exceptional orbits, and that $\mathrm {P} \sqcup \mathrm {Sing}(v) = \{ x \in M \mid \omega (x) \cup \alpha (x) \subseteq \mathrm {Sing}(v) \}$, 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 $\overline {\mathrm {LD}\sqcup \mathrm {Per}(v)} \supseteq M - \mathrm {Sing}(v)$, 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 $\mathrm {int}(\mathrm {Per}(v) \sqcup \mathrm {Sing}(v)) = \emptyset$ and $M - (\mathrm {P} \sqcup \mathrm {Sing}(v))$ 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 $\overline {\mathrm {P}} = \overline {\mathrm {LD}} =\mathbb {T}^2$.