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A theorem of Šarkovskii characterizing continuous maps of zero topological entropy. (English) Zbl 0686.26002
Let f be a continuous map of the compact real interval I into itself, \(f_ n\) the n-th iterate of f. For any \(x\in I\) \(\omega_ f(x)\) is the limit set of the sequence \(\{f^ n(x)\}^{\infty}_{n=1}.\)
The main aim of the paper is to give a new and short proof of the following result by A. N. Šarkovskii: f has a periodic point of period different from \(2^ n\) for any n if and only if there exists \(x\in I\) such that \(\omega_ f(x)\) is infinite and contains a periodic point.
Reviewer: K.Janková
MSC:
26A18 Iteration of real functions in one variable
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:
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