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Transactions of the American Mathematical Society

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Compact groups of homeomorphisms on tree-like continua

Authors: J. B. Fugate and T. B. McLean
Journal: Trans. Amer. Math. Soc. 267 (1981), 609-620
MSC: Primary 54F50; Secondary 54F20, 54H25
MathSciNet review: 626493
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Abstract: This paper is concerned with the fixed point sets of certain collections of homeomorphisms on a tree-like continuum. Extending a theorem of P. A. Smith, the authors prove that a periodic homeomorphism has a (nonvoid) continuum as its fixed point set. They then deduce possible periods for homeomorphisms on tree-like continua which satisfy certain decomposability or irreducibility conditions. The main result of the paper is that a compact group of homeomorphisms has a continuum as its fixed point set. This is applied to isometries. The paper concludes with sufficient conditions that a pointwise periodic homeomorphism have a fixed point.

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Keywords: Tree-like continuum, fixed point set, periodic homeomorphism, isometry, pointwise periodic homeomorphism, compact group of homeomorphisms, universal covering space, inverse limit system
Article copyright: © Copyright 1981 American Mathematical Society

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