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 there are no exceptional orbits and $\mathrm{P}\bigsqcup \mathrm{Sing}(v)=\{x\in M\mid \omega (x)\cup \alpha (x)\subseteq \mathrm{Sing}(v)\}$, where $\mathrm{P}$ is the union of non-closed proper orbits and $\bigsqcup$ is the disjoint union symbol. Moreover, $v$ is non-wandering if and only if $\overline{\mathrm{LD}\bigsqcup \mathrm{Per}(v)}\supseteq M-\mathrm{Sing}(v)$, 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 $\mathrm{int}(\mathrm{Per}(v)\bigsqcup \mathrm{Sing}(v))=\varnothing$ and $M-(\mathrm{P}\bigsqcup \mathrm{Sing}(v))$ 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 $\overline{\mathrm{P}}=\overline{\mathrm{LD}}={\mathbb{T}}^2$.

Cited by in corpus (2)

• Extended orbit properties on surfaces
• Recurrent and Non-wandering properties for foliations