Compact groups of homeomorphisms on tree-like continua

Fugate, J. B.; McLean, T. B.

doi:10.1090/S0002-9947-1981-0626493-5

Compact groups of homeomorphisms on tree-like continua
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by J. B. Fugate and T. B. McLean PDF

Trans. Amer. Math. Soc. 267 (1981), 609-620 Request permission

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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