Computing periods of powers (Q2897026)
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scientific article; zbMATH DE number 6053436
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Computing periods of powers |
scientific article; zbMATH DE number 6053436 |
Statements
5 July 2012
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discrete dynamical system
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set of periods
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interval maps
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circle maps
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iterates
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Sharkovsky's theorem
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Computing periods of powers (English)
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Given a continuous map \(f\) of the interval or of the circle, with a known set \(\mathrm{Per}(f)\) of minimal periods, the authors determine the set of minimal periods of an arbitrary iterate \(f^p\) of the map. More precisely, using a result from [\textit{L. Alsedà }, \textit{J. Llibre} and \textit{M. Misiurewicz}, Combinatorial dynamics and entropy in dimension one. Singapore: World Scientific (2000; Zbl 0963.37001)] it is proved that NEWLINE\[NEWLINE\mathrm{Per}(f^p) = \left\{\frac{k}{\mathrm{gcd}(k,p)}: k\in \mathrm{Per}(f)\right\}.NEWLINE\]
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