M-Shadowing and Transitivity for Flows

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 M-0-shadowing property or the M-1/2-shadowing property, then it is chain transitive. In addition, we prove that a Lyapunov stable flow with the M-0-shadowing or the M-1/2-shadowing is topologically transitive. Furthermore, it also is a minimal flow. As an application, we obtain that a C-1 generic vector field (X) over cap of a closed smooth 3-dimensional manifold with Sing((X) over cap) = empty set is Anosov provided that it has the M-0-shadowing property or the M-1/2-shadowing property.