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 | References | Similar Articles | Additional Information

Abstract: Let be continuous, and let denote the *p*th iterate of */*. Li and Yorke [2] proved that if there is a point such that but , 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 we construct a continuous, piecewise linear function such that *f* is chaotic, but almost every point of has eventual period *p*. The condition ``eventual period *p*'' cannot be replaced by ``period *p*". We prove that if for almost all , then for all . Moreover, we describe a normal form for all such ``square roots of the identity."

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Additional Information

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