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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Fixed points of arc-component-preserving maps


Author: Charles L. Hagopian
Journal: Trans. Amer. Math. Soc. 306 (1988), 411-420
MSC: Primary 54F20; Secondary 54H25
MathSciNet review: 927698
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Abstract: The following classical problem remains unsolved:

If $ M$ is a plane continuum that does not separate the plane and $ f$ is a map of $ M$ into $ M$, must $ f$ have a fixed point?

We prove that the answer is yes if $ f$ maps each arc-component of $ M$ into itself. Since every deformation of a space preserves its arc-components, this result establishes the fixed-point property for deformations of nonseparating plane continua. It also generalizes the author's theorem [10] that every arcwise connected nonseparating plane continuum has the fixed-point property. Our proof shows that every arc-component-preserving map of an indecomposable plane continuum has a fixed point. We also prove that every tree-like continuum that does not contain uncountably many disjoint triods has the fixed-point property for arc-component-preserving maps.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0927698-2
Keywords: Fixed-point property, deformation, arc-component-preserving map, nonseparating plane continua, indecomposable continua, internal composant, tree-like continua, uncountably many disjoint triods, free chain, Borsuk ray, dog-chases-rabbit principle
Article copyright: © Copyright 1988 American Mathematical Society