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Computing periods of powers. (English) Zbl 1247.37033
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 [L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one. Singapore: World Scientific (2000; Zbl 0963.37001)] it is proved that \[ \mathrm{Per}(f^p) = \left\{\frac{k}{\mathrm{gcd}(k,p)}: k\in \mathrm{Per}(f)\right\}. \]

MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37E10 Dynamical systems involving maps of the circle
37B20 Notions of recurrence and recurrent behavior in dynamical systems
37E99 Low-dimensional dynamical systems
26A18 Iteration of real functions in one variable
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