<abstract xmlns="http://www.w3.org/1999/xhtml">We present three different types of bijective functions <italic>f</italic> : <italic>I → I</italic> on a compact interval <italic>I</italic> 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 <italic>F</italic> : <italic>J → J</italic> of the identity of order <italic>k</italic> on a compact interval <italic>J</italic> with finitely many discontinuities is conjugate to a function <italic>f</italic> of type III, i.e., <italic>F</italic> = <italic>φ−</italic>1 ∘ <italic>f</italic> ∘ <italic>φ</italic> where <italic>φ</italic> is a continuous, bijective, and increasing mapping between <italic>J</italic> and [0, <italic>n</italic>] for some integer <italic>n</italic>.</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.

On Iteration of Bijective Functions with Discontinuities

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"On Iteration of Bijective Functions with Discontinuities"}

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...

On Iteration of Bijective Functions with Discontinuities

Published Online: Jul 09, 2020

Page range: 51 - 72

Received: Dec 23, 2019

Accepted: Jun 03, 2020

DOI: https://doi.org/10.2478/amsil-2020-0009

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

© 2020 Harald Fripertinger, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

On Iteration of Bijective Functions with Discontinuities

Published Online: Jul 09, 2020

Page range: 51 - 72

Received: Dec 23, 2019

Accepted: Jun 03, 2020

DOI: https://doi.org/10.2478/amsil-2020-0009

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

© 2020 Harald Fripertinger, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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