On representations of selfmappings

It is shown in this note that every “mild” self mapping $f:X \to X$ of a compact Hausdorff space $X$ into itself can be represented by the product $(Y,g) \times (Z,h)$ of two self mappings $g$ and $h$, where $g$ is a contraction $(\bigcap \nolimits _1^\infty {{g^n}(Y) = {\text {singleton}}} )$ and $h$ is a homeomorphism of $Z$ onto itself. Endowing the set of all selfmappings ${X^X}$ with the compact-open topology, the qualifier “mild” means that the closure of the family $\{ {f^n}|n \geqq 1\} \subset {X^X}$ is compact. In case $X$ is metrizable, some results of M. Edelstein and J. de Groot are used to linearize $(X,f)$ in the separable Hilbert space.