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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ \Gamma $-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
MathSciNet review: 0280655
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Abstract: We characterize the $ \Gamma $-compact continuous functions $ f:X \to X$ where $ X$ is a possibly-noncompact interval. The map $ f$ is called $ \Gamma $-compact if the closed topological semigroup $ \Gamma (f)$ generated by $ f$ is compact, or equivalently, if every sequence of iterates of $ f$ under functional composition $ (\ast)$ has a subsequence which converges uniformly on compact subsets of $ X$. For compact $ X$ the characterization is that the set of fixed points of $ f\ast f$ is connected. If $ X$ is noncompact an additional technical condition is necessary. We also characterize those maps $ f$ for which iterates of distinct orders agree ( $ \Gamma (f)$ finite) and state a result on common fixed points of commuting functions when one of the functions is $ \Gamma $-compact.

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Keywords: $ \Gamma $-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