No division implies chaos

Li, Tien Yien; Misiurewicz, Michał; Pianigiani, Giulio; Yorke, James A.

doi:10.1090/S0002-9947-1982-0664037-3

No division implies chaos
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by Tien Yien Li, Michał Misiurewicz, Giulio Pianigiani and James A. Yorke

Trans. Amer. Math. Soc. 273 (1982), 191-199

DOI: https://doi.org/10.1090/S0002-9947-1982-0664037-3

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