On Singular Interval-Valued Iteration Groups

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.