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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:
 3.7e+100 Low-dimensional dynamical systems
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