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Intervals containing all the periodic points. (English) Zbl 1385.37024

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be continuous and let Fix\((f^2)\) denote the set of all fixed points of \(f^2=f \circ f\). The authors prove that \(f^2\) of the convex hull of Fix\((f^2)\) contains \(P(f)\), the set of all periodic points of \(f\). They also study conditions under which \(f\) of the convex hull of Fix\((f^2)\) contains \(P(f)\). Several examples are given, including a real map \(f\) such that \(P(f) \not\subseteq f({\text{convex hull of Fix}}(f^2)).\)

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
54C30 Real-valued functions in general topology
37E05 Dynamical systems involving maps of the interval
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Full Text: DOI Euclid