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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Fixed points of arc-component-preserving maps
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by Charles L. Hagopian PDF
Trans. Amer. Math. Soc. 306 (1988), 411-420 Request permission

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.
References
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 306 (1988), 411-420
  • MSC: Primary 54F20; Secondary 54H25
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0927698-2
  • MathSciNet review: 927698