Abstract
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.
Highlights
During the ISFE54 Zygfryd Kominek raised discussion about the behavior of iterates of real functions with discontinuities
In the present paper three different types of bijective functions defined on a compact interval with finitely many removable and/or jump discontinuities will be presented, where certain iterates of these functions will be continuous
We will see that these examples of bijective functions are strongly related to permutations of finite sets
Summary
During the ISFE54 Zygfryd Kominek raised discussion about the behavior of iterates of real functions with discontinuities. In the present paper three different types of bijective functions defined on a compact interval with finitely many removable and/or jump discontinuities will be presented, where certain iterates of these functions will be continuous. We will see that these examples of bijective functions are strongly related to permutations of finite sets. We consider these functions as discrete structures, and in addition to analyzing their properties we will try to enumerate them. This way we obtain an overview on how many different types of these functions can be constructed
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