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

 

A generalization of a theorem of Janos and Edelstein


Author: Sehie Park
Journal: Proc. Amer. Math. Soc. 66 (1977), 344-346
MSC: Primary 54C25
MathSciNet review: 0454907
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Abstract: A generalization of Edelstein's version of a theorem of Janos and its converse are obtained:

Theorem. Let X be a compact metrizable topological space, and f, g be continuous self-maps of X such that $ gf = fg$ and f is bijective. Then g is injective and $ \bigcap\nolimits_1^\infty {{g^n}X = \{ {x_0}\} } $, where $ {x_0} \in X$, iff, given $ \lambda ,0 < \lambda < 1$, a homeomorphism h of X into $ {l_2}$ exists such that

$\displaystyle \left\Vert {h(gx) - h(gy)} \right\Vert = \lambda \left\Vert {h(fx) - h(fy)} \right\Vert$

for all $ x,y \in X$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0454907-7
PII: S 0002-9939(1977)0454907-7
Article copyright: © Copyright 1977 American Mathematical Society