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Common fixed-points for equicontinuous semigroups of mappings

Author: Theodore Mitchell
Journal: Proc. Amer. Math. Soc. 33 (1972), 146-150
MSC: Primary 26.54
MathSciNet review: 0289735
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Abstract: Let S be a semigroup of equicontinuous self maps of X, a compact Hausdorff space. It is shown that if S is left reversible (that is every pair of right ideals of S has nonempty intersection), then there is a compact group G of homeomorphisms of a retract Y of X with the property that S has a common fixed-point in X if and only if G has a common fixed-point in Y. As an application, it is proved that if F is a family of continuous commuting self maps of the closed unit interval I with the property that for each $ f \in F$, with one possible exception, the set of all iterates of f is equicontinuous, then I contains a common fixed-point of F.

References [Enhancements On Off] (What's this?)

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Keywords: Common fixed-points, equicontinuous mappings, left reversible semigroup, compact topological semigroups, commuting real functions
Article copyright: © Copyright 1972 American Mathematical Society

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