-compact maps on an interval and fixed points

Author:
William M. Boyce

Journal:
Trans. Amer. Math. Soc. **160** (1971), 87-102

MSC:
Primary 26.54; Secondary 22.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0280655-1

MathSciNet review:
0280655

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Abstract: We characterize the -compact continuous functions where is a possibly-noncompact interval. The map is called -compact if the closed topological semigroup generated by is compact, or equivalently, if every sequence of iterates of under functional composition has a subsequence which converges uniformly on compact subsets of . For compact the characterization is that the set of fixed points of is connected. If is noncompact an additional technical condition is necessary. We also characterize those maps for which iterates of distinct orders agree ( finite) and state a result on common fixed points of commuting functions when one of the functions is -compact.

**[1]**H. D. Block and H. P. Thielman,*Commutative polynomials*, Quart. J. Math. Oxford Ser. (2)**2**(1951), 241-243. MR**13**, 552. MR**0045250 (13:552f)****[2]**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)****[3]**Ralph DeMarr,*A common fixed point theorem for commuting mappings*, Amer. Math. Monthly**70**(1963), 535-537. MR**28**#2531. MR**0159314 (28:2531)****[4]**James Dugundji,*Topology*, Allyn and Bacon, Boston, Mass., 1966. MR**33**#1824. MR**0193606 (33:1824)****[5]**J. H. Folkman,*On functions that commute with full functions*, Proc. Amer. Math. Soc.**17**(1966), 383-386. MR**32**#8326. MR**0190916 (32:8326)****[6]**K. H. Hofmann and P. S. Mostert,*Elements of compact semigroups*, Merrill, Columbus, Ohio, 1966. MR**35**#285. MR**0209387 (35:285)****[7]**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)****[8]**J. T. Joichi,*On functions that commute with full functions and common fixed points*, Nieuw Arch. Wisk. (3)**14**(1966), 247-251. MR**34**#5078. MR**0205245 (34:5078)****[9]**J. L. Kelley,*General topology*, Van Nostrand, Princeton, N. J., 1955. MR**16**, 1136. MR**0070144 (16:1136c)****[10]**Theodore Mitchell,*Common fixed points for equicontinuous semigroups of mappings*, Notices Amer. Math. Soc.**16**(1969), 115. Abstract #663-101.**[11]**J. F. Ritt,*Permutable rational functions*, Trans. Amer. Math. Soc.**25**(1923), 399-448. MR**1501252****[12]**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)****[13]**A. D. Wallace,*The structure of topological semigroups*, Bull. Amer. Math. Soc.**61**(1955), 95-112. MR**16**, 796. MR**0067907 (16:796d)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0280655-1

Keywords:
-compact,
fixed point,
functional composition,
topological semigroup,
convergence of iteration,
commuting functions,
common fixed point,
precompact,
equicontinuous,
real functions

Article copyright:
© Copyright 1971
American Mathematical Society