Abstract
Let f:M→M be a diffeomorphism on a C ∞ n-dimensional manifold. Let C f (p) be the chain component of f associated to a hyperbolic periodic point p. In this paper, we show that (i) if f has the C 1 -stably orbitally shadowing property on the chain recurrent set R(f), then f satisfies both Axiom A and no-cycle condition, and (ii) if f has the C 1 -stably orbitally shadowing property on C f (p), then C f (p) is hyperbolic.MSC:37C50, 34D10, 37C20, 37C29.
Highlights
1 Introduction Let M be a closed C∞ n-dimensional manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C -topology
We say that f has the shadowing property on if for every >, there is δ > such that for any δ-pseudo-orbit {xi}bi=a ⊂ of f (–∞ ≤ a < b ≤ ∞), there is a point y ∈ M such that d(f i(y), xi) < for all a ≤ i ≤ b
It is easy to see that f has the shadowing property on if and only if f n has the shadowing property on for n ∈ Z \{ }
Summary
Let M be a closed C∞ n-dimensional manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C -topology. Of (y) ⊂ B (ξ ) and ξ ⊂ B Of (y) , where B (A) denotes the -neighborhood of a set A ⊂ M. f is said to have the weak shadowing property on (or is weakly shadowable) if for any > , there exists δ > such that for any δ-pseudo-orbit ξ = {xi}i∈Z ⊂ , there is a point y ∈ M such that ξ ⊂ B (Of (y)). In [ ], the authors showed that if f has the C -stably orbitally shadowing property, f satisfies both Axiom A and the strong transversality condition.
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