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A generalization of a theorem of Janos and Edelstein


Author: Sehie Park
Journal: Proc. Amer. Math. Soc. 66 (1977), 344-346
MSC: Primary 54C25
DOI: https://doi.org/10.1090/S0002-9939-1977-0454907-7
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$.

References [Enhancements On Off] (What's this?)

  • [1] M. Edelstein, A short proof of a theorem of L. Janos, Proc. Amer. Math. Soc. 20 (1969), 509-510. MR 38 #2743. MR 0234426 (38:2743)
  • [2] L. Janos, Converse of the Banach theorem in the case of one-to-one contracting mapping, Notices Amer. Math. Soc. 11 (1964), 686. Abstract #64T-469.
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DOI: https://doi.org/10.1090/S0002-9939-1977-0454907-7
Article copyright: © Copyright 1977 American Mathematical Society

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