Kinematic N-expansive flows

In light of the rich results of expansiveness in the dynamics of diffeomorphisms, it is natural to consider another notions of expansiveness such as countably-expansive, measure expansive, $N$-expansive and so on. In this paper, we introduce the notion of $N$-expansiveness for flows on a $C^{\infty}$ compact connected Riemannian manifold by using the kinematic expansiveness which is extension of the $N$-expansive diffeomorphisms. And we prove that a vector field $X$ on $M$ is $C^1$ robustly kinematic $N$-expansive then $X$ satisfies quasi-Anosov. Furthermore, we consider the hyperbolicity of local dynamical systems with kinematic $N$-expansiveness.