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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0542096-1

MathSciNet review:
542096

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Abstract | References | Similar Articles | Additional Information

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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*Edelstein’s contractivity and attractors*, Proc. Amer. Math. Soc. (to appear).

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Keywords:
Contractification,
nonexpansive,
contractive,
equicontinuous,
evenly continuous,
one point compactification

Article copyright:
© Copyright 1979
American Mathematical Society