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Shadowing property for inverse limit spaces. (English) Zbl 0762.58017

A continuous selfmap \(f\) of a compact metric space has the asymptotical shadowing property if for every \(\varepsilon>0\) there exists \(\delta>0\) and an integer \(N\geq 0\) such that every \(\delta\)-pseudo-orbit of \(f\) is \(\varepsilon\)-shadowed from the \(N\)-th term by an actual orbit of \(f\). If \(N=0\) we have the usual shadowing property. Two other related conditions are obtained by replacing the map \(f\) by the corresponding shift map on the inverse limit space.
The authors prove that if \(f\) is onto then all the four conditions are equivalent. As an application they show that the shift map of some pseudoarc has the shadowing property.

MSC:

37B99 Topological dynamics
26A18 Iteration of real functions in one variable
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