On representations of selfmappings
Author:
Ludvík Janoš
Journal:
Proc. Amer. Math. Soc. 26 (1970), 529-533
MSC:
Primary 54.60
DOI:
https://doi.org/10.1090/S0002-9939-1970-0270346-X
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 $\{ {f^n}|n \geqq 1\} \subset {X^X}$ 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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- A. D. Wallace, The Gebietstreue in semigroups, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 271–274. MR 0079008
- J. de Groot, Linearization of mappings, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 191–193. MR 0145004 ---, Every continuous mapping is linear, Notices Amer. Math. Soc. 6 (1959), 754. Abstract #560-65.
- Michael Edelstein, On the representation of mappings of compact metrizable spaces as restrictions of linear transformations, Canadian J. Math. 22 (1970), 372–375. MR 263040, DOI https://doi.org/10.4153/CJM-1970-045-1
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Keywords:
Representation,
self map,
mild self map,
squeezing self map,
Wallace “Swelling Lemma"
Article copyright:
© Copyright 1970
American Mathematical Society