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

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