AUTHOR=Zdun Marek C. 

TITLE=On Singular Interval-Valued Iteration Groups

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=2

YEAR=2016

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2016.00013

DOI=10.3389/fams.2016.00013

ISSN=2297-4687

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