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We show that if a C¹ generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C¹ generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles. [ABSTRACT FROM AUTHOR]

Let f : M → M be a diffeomorphism of a finite dimension, smooth compact Riemannian manifold M. In this paper, we demonstrate that if a diffeomorphism f lies within the C 1 interior of the set of all chain recurrence class-topologically stable diffeomorphisms, then the chain recurrence class is hyperbolic. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a generically C 1 vector field X of a compact smooth manifold M, if a homoclinic class H (γ , X) which contains a hyperbolic closed orbit γ is measure expansive for X then H (γ , X) is hyperbolic. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that if a differentiable map f of a compact smooth Riemannian manifold M is C 1 robustly positive continuum-wise expansive, then f is expanding. Moreover, C 1 -generically, if a differentiable map f of a compact smooth Riemannian manifold M is positively continuum-wise expansive, then f is expanding. [ABSTRACT FROM AUTHOR]