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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Stability of typical continuous functions with respect to some properties of their iterates
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by J. Smítal and K. Neubrunnová
Proc. Amer. Math. Soc. 90 (1984), 321-324
DOI: https://doi.org/10.1090/S0002-9939-1984-0727258-4

Abstract:

Let $I$ be a real compact interval, and let $C$ be the space of continuous functions $I \to I$ with the uniform metric. For $f \in C$ denote $\nu (f) = {\sup _{x \in I}}(\lim {\sup _{n \to x}}{f^n}(x) - \lim {\inf _{n \to x}}{f^n}(x))$, where ${f^n}$ is the $n$th iterate of $f$. Then for each positive $d$ there is an open set ${C^*}$ dense in $C$ such that the oscillation of $v$ at each point of ${C^*}$ is less than $d$. Consequently, $\nu$ is continuous in $C$ except of the points of a first Baire category set.
References
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Bibliographic Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 321-324
  • MSC: Primary 54H20; Secondary 26A18
  • DOI: https://doi.org/10.1090/S0002-9939-1984-0727258-4
  • MathSciNet review: 727258