On iteration of bijective functions with discontinuities

Harald Fripertinger

Published: 2020-07-09

Vol. 34 No. 1 (2020)

On iteration of bijective functions with discontinuities

Harald Fripertinger

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◦f◦ϕ where ϕ is a continuous, bijective, and increasing mapping between J and [0,n] for some integer n.

Keywords:

iteration of functions , bijective functions on a compact interval , derangements , permutations with successions , wreath products , iterative roots of the identity , enumeration

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Fripertinger, H. (2020). On iteration of bijective functions with discontinuities. Annales Mathematicae Silesianae, 34(1), 51–72. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13632

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Vol. 34 No. 1 (2020)
Published: 2020-07-20

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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