A generalization of a theorem of Janos and Edelstein
HTML articles powered by AMS MathViewer
- by Sehie Park PDF
- Proc. Amer. Math. Soc. 66 (1977), 344-346 Request permission
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 \[ \left \| {h(gx) - h(gy)} \right \| = \lambda \left \| {h(fx) - h(fy)} \right \|\] for all $x,y \in X$.References
- Michael Edelstein, A short proof of a theorem of L. Janos, Proc. Amer. Math. Soc. 20 (1969), 509–510. MR 234426, DOI 10.1090/S0002-9939-1969-0234426-9 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. —, Homothetic property of contractive one-to-one mappings, Notices Amer. Math. Soc. 13 (1966), 818. Abstract #638-11. —, One-to-one contractive mappings on compact space, Notices Amer. Math. Soc. 14 (1967), 133. Abstract #67T-21.
- Gerald Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), no. 4, 261–263. MR 400196, DOI 10.2307/2318216
Additional Information
- © Copyright 1977 American Mathematical Society
- 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