On Singular Interval-Valued Iteration Groups - Zdun, Marek C. (Frontiers in Applied Mathematics and Statistics, 2016)
Let I=(a,b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ:I→ R be a non-increasing continuous surjection constant on the components of IL and the closures of these components be the maximal intervals of constancy of φ. The family {F^t,tR} of the interval-valued functions F^t(x):=φ^-1[t+φ(x)], xI forms a set-valued iteration group. We determine a maximal dense subgroup TR such that the set-valued subgroup {F^t,tT} has some regular properties. In particular, the mappings Tt→ F^t(x) for tT possess selections f^t(x) F^t(x), which are disjoint group of continuous functions.