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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

No division implies chaos


Authors: Tien Yien Li, Michał Misiurewicz, Giulio Pianigiani and James A. Yorke
Journal: Trans. Amer. Math. Soc. 273 (1982), 191-199
MSC: Primary 28D20; Secondary 58F13
MathSciNet review: 664037
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Abstract: Let $ I$ be a closed interval in $ {R^1}$ and $ f:I \to I$ be continuous. Let $ {x_0} \in I$ and

$\displaystyle {x_{i + 1}} = f({x_i})\quad {\text{for}}\;i > 0.$

We say there is no division for $ ({x_1},{x_2}, \ldots ,{x_n})$ if there is no $ a \in I$ such that $ {x_j} < a$ for all $ j$ even and $ {x_j} < a$ for all $ j$ odd. The main result of this paper proves the simple statement: no division implies chaos.

Also given here are some converse theorems, detailed estimates of the existing periods, and examples which show that, under our conditions, one cannot do any better.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0664037-3
PII: S 0002-9947(1982)0664037-3
Keywords: No division, periodic orbit, chaos, entropies
Article copyright: © Copyright 1982 American Mathematical Society