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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
DOI: https://doi.org/10.1090/S0002-9939-1976-0417351-3
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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DOI: https://doi.org/10.1090/S0002-9939-1976-0417351-3
Keywords: Iterations, nonlinear functions, chaotic functions, dynamical systems
Article copyright: © Copyright 1976 American Mathematical Society