Abstract
Let f be a diffeomorphism on a closed smooth manifold M. In this paper, we show that f has the -stably limit shadowing property on the chain component of f containing a hyperbolic periodic point p, if and only if is a hyperbolic basic set. MSC:37C50, 34D10.
Highlights
Various closed invariant sets in dynamical systems are natural candidates to replace Smale’s hyperbolic basic sets in non-hyperbolic theory of differentiable dynamical systems see [ – ]).To investigate the above, we deal with the shadowing property
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
We introduce the limit shadowing property which was introduced and studied by
Summary
Various closed invariant sets (transitive set, chain transitive set, homoclinic class, chain component, etc.) in dynamical systems are natural candidates to replace Smale’s hyperbolic basic sets in non-hyperbolic theory of differentiable dynamical systems see [ – ]).To investigate the above, we deal with the shadowing property. For a closed f -invariant set ⊂ M, we say that is chain transitive if for any point x, y ∈ and δ > , there exists a δ-pseudo orbit {xi}bi=δaδ ⊂ (aδ < bδ) of f such that xaδ = x and xbδ = y. 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.
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