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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Contractification of a semigroup of maps

Authors: Hwei Mei Ko and Kok Keong Tan
Journal: Proc. Amer. Math. Soc. 77 (1979), 267-275
MSC: Primary 54E10
MathSciNet review: 542096
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Abstract: Let $ (X,\tau )$ be a metrizable topological space, $ \mathcal{P}(\tau )$ be the family of all metrics on X whose metric topologies are $ \tau $. Assume that the semigroup F of maps from X into itself, with composition as its semigroup operation, is equicontinuous under some $ d \in \mathcal{P}(\tau )$; then we have the following results:

I. There exists $ d' \in \mathcal{P}(\tau )$ such that f is nonexpansive under $ d'$ for each $ f \in F$.

II. If F is countable, commutative, and for each $ f \in F$, there is $ {x_f} \in X$ such that the sequence $ ({f^n}(x))_{n = 1}^\infty $ converges to $ {x_f},\forall x \in X$, then there exists $ d'' \in \mathcal{P}(\tau )$ such that f is contractive under $ d''$ for each $ f \in F$.

III. If there is $ p \in X$ such that (1) $ {\lim _{n \to \infty }}{f^n}(x) = p,\forall x \in X$ and $ \forall f \in F$, (2) there is a neighbourhood B of p such that $ {\lim _{m \to \infty }}{f_{{n_1}}}{f_{{n_2}}} \cdots {f_{{n_m}}}(B) = \{ p\} $ for any choice of $ {f_{{n_i}}} \in F,i = 1, \ldots ,m$, and the limit depends on m only, then for each $ \lambda $ with $ 0 < \lambda < 1$, there exists $ d''' \in \mathcal{P}(\tau )$ such that each f in F is a Banach contraction under $ d'''$ with Lipschitz constant $ \lambda $.

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Keywords: Contractification, nonexpansive, contractive, equicontinuous, evenly continuous, one point compactification
Article copyright: © Copyright 1979 American Mathematical Society

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