Kinematic N-expansive flows

TL;DR

This paper introduces the concept of $N$-expansiveness for flows on smooth manifolds, extending the idea of kinematic expansiveness, and explores its implications for robustness and hyperbolicity in dynamical systems.

Contribution

It defines $N$-expansiveness for flows using kinematic expansiveness and establishes its connection to quasi-Anosov systems and hyperbolic behavior.

Findings

Robust $C^1$ kinematic $N$-expansive vector fields are quasi-Anosov.

Kinematic $N$-expansiveness relates to hyperbolic properties of local dynamical systems.

Extension of $N$-expansiveness concepts from diffeomorphisms to flows.

Abstract

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.