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Diffeomorphisms with weak shadowing

100%

Sakai K.

Fundamenta Mathematicae

2001

tom 168

nr 1

57-75

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 $C^{∞}$ 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.

C¹-Stably Positively Expansive Maps

100%

Sakai K.

nr 2

197-209

The notion of C¹-stably positively expansive differentiable maps on closed $C^∞$ 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.