On representations of selfmappings

Abstract

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