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On a converse of Sharkovsky’s theorem. (English) Zbl 0893.58024
A. N. Sharkovskij [Ukr. Math. Zh. 16, 61-71 (1964; Zbl 0122.17504)] proved the following theorem on periodic points of continuous maps on intervals.
Theorem (Sharkovskij). Let \(f: I\to I\) be a continuous map from the interval \(I\) into itself. If positive integers \(k\) and \(r\) obey Sharkovskij’s ordering, i.e., \(k\triangleright r\) and \(f\) has a point of period \(k\), then \(f\) must have a point of period \(r\).
The question addressed in the paper is the following: Given any positive integers \(k\), \(r\) with \(k\triangleright r\), is there a continuous map that has a point of period \(r\) but no points of period \(k\)? As an answer to this question the author obtains the following result.
Theorem. For any positive integer \(r\) there exists a continuous map \(f_r: I_r \to I_r\) on the interval \(I_r\) such that \(f_r\) has points of prime period \(r\) but no points of prime period \(s\) for all positive integers \(s\) that precede \(r\) in the Sharkovskij’s ordering, i.e., \(s \triangleright \cdots \triangleright r\).

MSC:
37E99 Low-dimensional dynamical systems
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