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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Piecewise linear functions with almost all points eventually periodic
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by Melvyn B. Nathanson PDF
Proc. Amer. Math. Soc. 60 (1976), 75-81 Request permission

Abstract:

Let $f:[0,1] \to [0,1]$ be continuous, and let ${f^p}$ denote the pth iterate of /. Li and Yorke [2] proved that if there is a point $x \in [0,1]$ such that ${f^3}(x) = x$ but $f(x) \ne x$, then f is chaotic in the sense that f has periodic points of arbitrarily large period, and uncountably many points which are not even asymptotically periodic. But this chaos can be measure theoretically trivial. For each $p \geqslant 3$ we construct a continuous, piecewise linear function $f:[0,1] \to [0,1]$ such that f is chaotic, but almost every point of $[0,1]$ has eventual period p. The condition “eventual period p” cannot be replaced by “period p". We prove that if ${f^p}(x) = x$ for almost all $x \in [0,1]$, then ${f^2}(x) = x$ for all $x \in [0,1]$. Moreover, we describe a normal form for all such “square roots of the identity."
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 75-81
  • MSC: Primary 26A18
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0417351-3
  • MathSciNet review: 0417351