On Singular Interval-Valued Iteration Groups

Marek C Zdun

doi:10.3389/fams.2016.00013

Abstract

Let $I=(a,b)$ and $L$ be a nowhere dense perfect set containing the ends of the interval $I$ and let $\varphi:I\to \mathbb{R}$ be a non-increasing continuous surjection constant on the components of $I\setminus L$ and the closures of these components be the maximal intervals of constancy of $\varphi$. The family $\{F^t,t\in \mathbb{R}\}$ of the interval-valued functions $F^t(x):=\varphi^{-1}[t+\varphi(x)]$, $x\in I$ forms a set-valued iteration group. We determine a maximal dense subgroup $T\subsetneq \mathbb{R}$ such that the set-valued subgroup $\{F^t,t\in T\}$ has some regular properties. In particular, the mappings $T\backepsilon t\to F^t(x)$ for $t\in T$ possess selections $f^t(x) \in F^t(x), $ which are disjoint group of continuous functions.

Highlights

A family of functions {f t : I → I, t ∈ R} such that f t ◦ f s = f t+s, t, s ∈ R is said to be an iteration group, a family of set-valued functions {Ft : I → 2I, t ∈ R} such that Ft ◦ Fs = Ft+s, t, s ∈ R is said to be a set-valued iteration group
The considered s-v iteration groups have the very irregular properties. For every such s-v iteration group {Ft : I → 2I, t ∈ R} we find a special maximal additive subgroup T ⊂ R such that group {Ft : I → 2I, t ∈ T} has several “regular” properties
It is easy to verify that the s-v iteration group {Ft : I → cc[I], t ∈ R} generated by φ has the following properties

Summary

Introduction

A family of functions {f t : I → I, t ∈ R} such that f t ◦ f s = f t+s, t, s ∈ R is said to be an iteration group, a family of set-valued functions {Ft : I → 2I, t ∈ R} such that Ft ◦ Fs = Ft+s , t, s ∈ R is said to be a set-valued iteration group (abbreviated to s-v iteration group). The notion of an iteration semigroup of set-valued functions was introduced and studied by Smajdor [1] and studied in some classes of set-valued functions (see e.g., [2], [3], [4], [5]). The considered s-v iteration groups have the very irregular properties. For every such s-v iteration group {Ft : I → 2I, t ∈ R} we find a special maximal additive subgroup T ⊂ R such that group {Ft : I → 2I, t ∈ T} has several “regular” properties

Objectives

Methods

Results

Conclusion