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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.

##### MSC:
 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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