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Proceedings of the American Mathematical Society

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



Piecewise linear functions with almost all points eventually periodic

Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 60 (1976), 75-81
MSC: Primary 26A18
MathSciNet review: 0417351
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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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Keywords: Iterations, nonlinear functions, chaotic functions, dynamical systems
Article copyright: © Copyright 1976 American Mathematical Society