On a functional equation by Baak, Boo and Rassias Publications, to be read Families of Commuting Formal Power Series and Formal Functional Equations On Iteration of Bijective Functions with Discontinuities

On Iteration of Bijective Functions with Discontinuities

Annales Mathematicae Silesianae 34 Issue 1, 51-72 (2020). DOI 10.2478/amsil-2020-0009

Abstract. We present three different types of bijective functions f: I→ I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F: J→ J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F=φ-1 o f o φ where φ is a continuous, bijective, and increasing mapping between J and [0,n] for some integer n.


harald.fripertinger "at" uni-graz.at, October 3, 2024

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