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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On representations of selfmappings

Author: Ludvík Janoš
Journal: Proc. Amer. Math. Soc. 26 (1970), 529-533
MSC: Primary 54.60
MathSciNet review: 0270346
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Abstract: 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}\vert 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.

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PII: S 0002-9939(1970)0270346-X
Keywords: Representation, self map, mild self map, squeezing self map, Wallace ``Swelling Lemma"
Article copyright: © Copyright 1970 American Mathematical Society