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Proceedings of the American Mathematical Society

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



Stability of typical continuous functions with respect to some properties of their iterates

Authors: J. Smítal and K. Neubrunnová
Journal: Proc. Amer. Math. Soc. 90 (1984), 321-324
MSC: Primary 54H20; Secondary 26A18
MathSciNet review: 727258
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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.

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Article copyright: © Copyright 1984 American Mathematical Society