Abstract
Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness for continuous dynamical systems on a compact connected smooth manifold M by using the dense set D of M, which is different from the weak expansive flows. We establish some examples having the countably expansive property, and we prove that if a vector field X of M is C 1 stably countably expansive then it is quasi-Anosov.
Highlights
Let X be a compact metric space with a metric d and f : X → X be a homeomorphism
The theory of dynamical systems is motivated by the search of knowledge about the orbits of a given dynamical systems
To describe the dynamics on the underlying space, it is usual to use the notion of expansiveness
Summary
Let X be a compact metric space with a metric d and f : X → X be a homeomorphism. Utz [1]. From the result of [4], Moriyasu, Sakai, and Sun [7] extended expansive diffeomorphisms to vector fields about the C1 stably point of view. That is, they showed that if a vector field X is C1 stably expansive it is quasi-Anosov. We prove that if a vector field X of a compact connected manifold M is C1 stably countably expansive, it is quasi-Anosov which is a general result of Moriyasu, Sakai, and. We have that if a vector field X of a compact connected manifold M is C1 stably expansive, weak expansive, and countably expansive it is quasi-Anosov
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