A note on periods of powers*

• Main Authors: Cánovas J.S., Linero Bas A.
• Format: Article
• Language: English
• Published: EDP Sciences 2014-11-01
• Series: ESAIM: Proceedings and Surveys
• Subjects: discrete dynamical systems; set of periods; iterates; sharkovsky’s theorem; cyclic permutation; direct product; σ-permutation map; sharkovsky type; functional equation
• Online Access: http://dx.doi.org/10.1051/proc/201446011

Let f:X → X be a continuous map defined from a topological space X into itself. We discuss the problem of analyzing and computing explicitly the set Per(fp) of periods of the p-th iterate fp from the knowledge of the set Per(f) of periods of f. In the case of interval or circle maps, that is, X = [0,1] or X = S1, this question was solved in [11]. Now, we present some remarks and advances concerning the set Per(fp) for a continuous interval map, and on the other hand we study and solve the problem when we consider σ-permutation maps, namely, when X = [0,1] k for some integer k ≥ 2 and the map has the form F(x1,x2,...,xk) = (fσ(1)(xσ(1)),fσ(2)(xσ(2)),...,fσ(k)(xσ(k))), being each fj a continuous interval map and σ a cyclic permutation of {1,2,...,k}. This paper can be seen as the continuation of [11].