Common fixed-points for equicontinuous semigroups of mappings

Mitchell, Theodore

doi:10.1090/S0002-9939-1972-0289735-4

Common fixed-points for equicontinuous semigroups of mappings
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Proc. Amer. Math. Soc. 33 (1972), 146-150 Request permission

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.

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