On Singular Interval-Valued Iteration Groups

Frontiers in Applied Mathematics and Statistics

Zdun, Marek C.¹ 

[1] Institute of Mathematics, Pedagogical University of Cracow, Kraków, Poland

关键词: iteration group;    Set-valued functions;    Simultaneous functional equations;    Cantor set;    singular Lebesgue function;   

DOI  :  10.3389/fams.2016.00013

学科分类：数学（综合）

来源: Frontiers

【 摘 要 】

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.

【 授权许可】

CC BY