A topological characterization for non-wandering surface flows

TL;DR

This paper characterizes non-wandering surface flows topologically, establishing conditions for equivalence to smooth flows without exceptional orbits, and explores the relationship between non-wandering, transitivity, and orbit closures on compact surfaces.

Contribution

It provides a new topological characterization of non-wandering surface flows, including conditions for equivalence to smooth flows and criteria for topological transitivity.

Findings

Non-wandering flows are equivalent to smooth flows without exceptional orbits.

Non-wandering condition characterized by closure of locally dense and periodic orbits.

Constructed smooth flow on torus with dense proper and locally dense orbits.

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 $\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v) = \{ x \in M \mid \omega(x) \cup \alpha(x) \subseteq \mathop{\mathrm{Sing}}(v) \}$, 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 $\overline{\mathrm{P}} = \overline{\mathrm{LD}} =\mathbb{T}^2$.