Common fixed-points for equicontinuous semigroups of mappings

Author:
Theodore Mitchell

Journal:
Proc. Amer. Math. Soc. **33** (1972), 146-150

MSC:
Primary 26.54

DOI:
https://doi.org/10.1090/S0002-9939-1972-0289735-4

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 , with one possible exception, the set of all iterates of *f* is equicontinuous, then *I* contains a common fixed-point of *F*.

**[1]**W. M. Boyce,*Commuting functions with no common fixed point*, Trans. Amer. Math. Soc.**137**(1969), 77-92. MR**38**#4627. MR**0236331 (38:4627)****[2]**-,*On*-*compact maps on an interval and commutativity*, Trans. Amer. Math. Soc.**160**(1971), 87-102. MR**0280655 (43:6374)****[3]**K. H. Hofmann and P. S. Mostert,*Elements of compact semigroups*, Merrill, Columbus, Ohio, 1966. MR**35**#285. MR**0209387 (35:285)****[4]**J. P. Huneke,*On common fixed points of commuting continuous functions on an interval*, Trans. Amer. Math. Soc.**139**(1969), 371-381. MR**38**#6005. MR**0237724 (38:6005)****[5]**J. L. Kelley,*General topology*, Van Nostrand, Princeton, N.J., 1955. MR**16**, 1136. MR**0070144 (16:1136c)****[6]**A. L. Shields,*On fixed points of commuting analytic functions*, Proc. Amer. Math. Soc.**15**(1964), 703-706. MR**29**#2790. MR**0165508 (29:2790)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0289735-4

Keywords:
Common fixed-points,
equicontinuous mappings,
left reversible semigroup,
compact topological semigroups,
commuting real functions

Article copyright:
© Copyright 1972
American Mathematical Society