Genericity for non-wandering surface flows

Abstract

Consider the set ${\chi }_{\mathrm{nw}}^0$ 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 ${\chi }_{\mathrm{nw}}^0$. 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 ${\mathbb{K}}^2$.