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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$\Gamma$-compact maps on an interval and fixed points
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by William M. Boyce PDF
Trans. Amer. Math. Soc. 160 (1971), 87-102 Request permission

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.
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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