On Singular Interval-Valued Iteration Groups

• Published in: Frontiers in applied mathematics and statistics Vol. 2
• Main Author: Zdun, Marek C.
• Format: Journal Article
• Language: English
• Published: Frontiers Media S.A 13.09.2016
• Subjects: Cantor set; iteration group; Set-valued functions; Simultaneous functional equations; singular Lebesgue function
• ISSN: 2297-4687; 2297-4687
• DOI: 10.3389/fams.2016.00013

Let I=(a,b) and L be a nowhere dense perfect set containing the ends of the interval I and let varphi:Ito mathbb R be a non-increasing continuoussurjection constant on the components of Isetminus L and the closures of these components be the maximal intervals of constancy of varphi . The family F^t,tin mathbb R of theinterval-valued functions F^t(x):=varphi^ -1 t+varphi(x) , xin I forms a set-valued iteration group. We determine a maximal densesubgroup Tsubsetneq mathbb R such that the set-valued subgroup F^t,tin T has some regular properties. In particular, the mappings Tbackepsilon tto F^t(x) for tin T possess selections f^t(x) in F^t(x), which are disjoint group of continuous functions.