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Chaotic map with the property of recurrence. (English) Zbl 1116.37007

Summary: We consider a continuous map \(f: X\rightarrow X\), where \(X\) is a compact metric space, and discuss the existence of a chaotic set of \(f\) specially (as \(X=[0,1]\)). We prove that \(f\) has a positively topological entropy if and only if it has an uncountably chaotic set in which each point is recurrent and is not weakly periodic.

MSC:

37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37E05 Dynamical systems involving maps of the interval
37B40 Topological entropy
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