Asymptotic average shadowing property on compact metric spaces. (English) Zbl 1152.54027

Honary, B.; Bahabadi, A. Zamani

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 9, 2857-2863 (2008).

Authors’ summary: We prove that if for a continuous map \(f\) on a compact metric space \(X\), the chain recurrent set, \(R(f)\) has more than one chain component, then \(f\) does not satisfy the asymptotic average shadowing property. We also show that if a continuous map \(f\) on a compact metric space \(X\) has the asymptotic average shadowing property and if \(A\) is an attractor for \(f\), then \(A\) is the single attractor for \(f\) and we have \(A=R(f)\). We also study diffeomorphisms with asymptotic average shadowing property and prove that if \(M\) is a compact manifold which is not finite with \(\dim M=2\), then the \(C^{1}\) interior of the set of all \(C^{1}\) diffeomorphisms with the asymptotic average shadowing property is characterized by the set of \(\Omega \)-stable diffeomorphisms.

Reviewer: Ludvík Janoš (Claremont)

MSC:

54H20	Topological dynamics (MSC2010)
54F99	Special properties of topological spaces

Keywords:

asymptotic average shadowing; \(\delta \)-average-pseudo-orbit; shadowing property; \(\delta \)-pseudo-orbit; chain component; chain recurrent; attractor

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