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On common fixed points, periodic points, and recurrent points of continuous functions. (English) Zbl 1037.54018
Let \(C(I,I)\) be the class of all self-mappings on \(I = [0,1]\). For a mapping \(f \in C(I,I)\) and a positive integer \(m\), denote by \(f^m\) the \(m\)-th iterate of \(f\), and let \(F_m(f) = \{x \in I: f^m(x) = x\}\), and \(\mathcal H = \{f, g: f, g \in C(I,I)\}\). The following results are shown.
1. A subset \(S\) of \(I\) is nowhere dense if and only if \(\{f \in C(I): F_m(f) \cap \overline{S} \neq \emptyset\}\) is a nowhere dense subset of \(C(I)\).
2. The set \(\{f, g \in \mathcal H: f \circ g = g \circ f\}\) is a nonempty, closed and nowhere dense subset of \(\mathcal H\).
3. Sufficient conditions are shown under which the set \(R(f)\) of recurrent points of \(f \in C(I,I)\) is a closed interval.
Other results about dynamics on \(I\) are presented.

54C50 Topology of special sets defined by functions
54C60 Set-valued maps in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
37E05 Dynamical systems involving maps of the interval
26A21 Classification of real functions; Baire classification of sets and functions
54H20 Topological dynamics (MSC2010)
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