Fixed point theorems for plane continua with applications
 Author/Creator
 Blokh, Alexander M., 1958
 Language
 English.
 Publication
 Providence, Rhode Island : American Mathematical Society, 2013.
 Physical description
 xiii, 97 pages : illustrations ; 26 cm.
 Series
 Memoirs of the American Mathematical Society ; number 1053.
Access
Available online

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QA3 .A57 NO.1053

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QA3 .A57 NO.1053
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Contributors
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 Introduction Part 1. Basic Theory: Preliminaries and outline of Part 1 Tools Partitions of domains in the sphere Part 2. Applications of Basic Theory: Description of main results of Part 2 Outchannels and their properties Fixed points Bibliography Index.
 (source: Nielsen Book Data)
 Publisher's Summary
 In this memoir the authors present proofs of basic results, including those developed so far by Harold Bell, for the plane fixed point problem: Does every map of a nonseparating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. The authors define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation 1. A prime end theory is developed through hyperbolic chords in maximal round balls contained in the complement of a nonseparating plane continuum $X$. They define the concept of an outchannel for a fixed point free map which carries the boundary of $X$ minimally into itself and prove that such a map has a unique outchannel, and that outchannel must have variation $1$. Also Bell's Linchpin Theorem for a foliation of a simply connected domain, by closed convex subsets, is extended to arbitrary domains in the sphere. The authors introduce the notion of an oriented map of the plane and show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982.
(source: Nielsen Book Data)
Subjects
 Subject
 Fixed point theory.
Bibliographic information
 Publication date
 2013
 Responsibility
 Alexander M. Blokh, Robbert J. Fokkink, John C. Mayer, Lex G. Oversteegen, E.D. Tymchatyn.
 Series
 Memoirs of the American Mathematical Society, 00659266 ; number 1053
 Note
 "July 2013, volume 224, number 1053 (second of 4 numbers)."
 ISBN
 9780821884881
 0821884883