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

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Summary

Introduction

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.

Results
Conclusion

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