Genericity for non-wandering surface flows

TL;DR

This paper studies non-wandering flows on closed surfaces, showing they can be approximated by regular flows without certain complex behaviors, and characterizes when such flows are topologically stable.

Contribution

It provides a characterization of topologically stable non-wandering flows on closed surfaces and demonstrates their non-existence on most surfaces except specific cases.

Findings

Flow approximation by regular non-wandering flows without heteroclinic connections

Topological stability characterized by orbit space homeomorphic to a closed interval

Non-existence of stable flows on surfaces other than S^2, P^2, and K^2

Abstract

Consider the set $\chi^0_{\mathrm{nw}}$ of non-wandering continuous flows on a closed surface. Then such a flow can be approximated by regular non-wandering flows without heteroclinic connections nor locally dense orbits in $\chi^0_{\mathrm{nw}}$. Using this approximation, we show that a non-wandering continuous flow on a closed connected surface is topologically stable if and only if the orbit space of it is homeomorphic to a closed interval. Moreover we state the non-existence of topologically stable non-wandering flows on closed surfaces which are not neither $\mathbb{S}^2$, $\mathbb{P}^2$, nor $\mathbb{K}^2$.