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 φ : I → R be a non-increasing continuous surjection constant on the components of I \ L and the closures of these components be the maximal intervals of constancy of . The family F t t R of the interval-valued functions F t 1 φ { , ∈ } (x): = φ [t + φ(x)], x ∈ I forms a set-valued iteration group. We determine a maximal dense subgroup T ( R such that the set-valued subgroup {F , t ∈ T} has some regular properties. In particular, the mappings T  t → F (x) for t ∈ T possess selections f (x) ∈ F (x), which are disjoint group of continuous functions.