In this paper, we show that the C interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the C interior of the set of all expansive diffeomorphisms. And the C interior of the set of all continuum-wise fully expansive diffeomorphisms on a surface is investigated. Furthermore, we have necessary and sufficient conditions for a diffeomorphism belonging to these open sets to be Anosov.
The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.
The notion of C¹-stably positively expansive differentiable maps on closed manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.
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