${\mathscr{M}}$-Shadowing and Transitivity for Flows

Abstract

Smale pointed out a very important problem in dynamical systems theory is to find the minimal set. In this paper, we show that if a flow on compact metric space has the ${\mathscr{M}}_{0}$-shadowing property or the ${\mathscr{M}}^{\frac {1}{2}}$-shadowing property, then it is chain transitive. In addition, we prove that a Lyapunov stable flow with the ${\mathscr{M}}_{0}$-shadowing or the ${\mathscr{M}}^{\frac {1}{2}}$-shadowing is topologically transitive. Furthermore, it also is a minimal flow. As an application, we obtain that a C¹ generic vector field $\hat {X}$ of a closed smooth 3-dimensional manifold with $\text{Sing}(\hat {X})=\varnothing$ is Anosov provided that it has the ${\mathscr{M}}_{0}$-shadowing property or the ${\mathscr{M}}^{\frac {1}{2}}$-shadowing property.