Genericity for Non-Wandering Surface Flows

Abstract

Consider the set źnw0$\chi ^{0}_{\text {nw}}$ of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement MźPer(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in źnw0$\chi ^{0}_{\text {nw}}$; (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in źnw0$\chi ^{0}_{\text {nw}}$ if and only if the surface is homeomorphic to either the sphere S2$\mathbb {S}^{2}$, the projective plane ź2$\mathbb {P}^{2}$, or the Klein bottle K2$\mathbb {K}^{2}$.