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Discrete Open and Closed Mappings on Generalized Continua and Newman's Property

Published online by Cambridge University Press:  20 November 2018

Louis F. McAuley  and
Eric E. Robinson
Louis F. McAuley
Affiliation:
The State University of New York at Binghamton, Binghamton, New York
Eric E. Robinson
Affiliation:
Ithaca College, Ithaca, New York
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In 1930, M. H. A. Newman proved a rather remarkable theorem which has become one of the classical theorems in topology. It has many important applications. A special case of Newman's Theorem is that a periodic homeomorphism of period n > 1 of a sphere S onto itself must have some orbit which is not contained in a “cap” smaller than a hemisphere. The general theorem is as follows:

THEOREM ([15]). Suppose that Mn is a connected (metric) n-manifold, U is a domain in Mn, andp is an integer greater than 1. Then there is a positive number d such that no uniformly continuous homeomorphism h of Mn onto itself of period p moves every point of U a distance < d. That is, there is x ∊ U so that the orbit of x under h has diameter ≦ d.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 1081 - 1112
Copyright
Copyright © Canadian Mathematical Society 1984

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