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On the average-shadowing property. (English) Zbl 1022.37017
Summary: It is proved that a homeomorphism on a compact metric space with the average-shadowing property has only one chain component which is the whole space. In particular, every point is chain recurrent. Using this result a short proof for a recent theorem of Sakai is given.

##### MSC:
 37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics 37B35 Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)