Iterated function systems enriched with symmetry
type: Item
instance of: scholarly article
title: Iterated function systems enriched with symmetry
zbMATH Open document ID: 1505.28009
author: Krzysztof Leśniak
author: Nina Snigireva
published in: Constructive Approximation
publication date: 2022-12-19T00:00:00Z
review text: This paper deals with finite IFSs (iterated function systems) which are contractive on average. ``The average contractivity of an IFS guarantees the asymptotic stability of the Markov operator associated with this IFS. This in turn ensures the existence and uniqueness of the invariant measure.'' One can cite authors' abstract: ``An iterated function system (IFS) can be enriched with an isometry in such a way that the resulting fractal set has prescribed symmetry. If the original system is contractive, then its associated self-similar set is an attractor. On the other hand, the enriched system is no longer contractive and therefore does not have an attractor. However, it posses a self-similar set which, under certain conditions, behaves like an attractor. We give a rigorous procedure which relates a given enriched IFS to a contractive one. Further, we link this procedure to the Lasota--Myjak theory of semiattractors, and so via invariant measures to probabilistic iterated function systems. The chaos game algorithm for enriched IFSs is discussed. We illustrate our main results with several examples which are related to classical fractals such as the Sierpiński triangle and the Barnsley fern.'' A brief survey of this research is devoted to the theory of contractive IFS and its extensions. Also, the attention is given to noncontractive IFSs. Connections between presented results and known results, as well as explanations of used techniques are given. Auxiliary notions and few known results are recalled. Several auxiliary statements are proven. Finally, some questions related to the further research in this topic are considered.
reviewed by: Symon Serbenyuk
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37B55
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 7634049
zbMATH Keywords: chaos game
zbMATH Keywords: isometry
zbMATH Keywords: iterated function system
zbMATH Keywords: semiattractor
describes a project that uses: Symmetry in Chaos
MaRDI profile type: Publication
full work available at URL: s00365-021-09560-3
OpenAlex ID: W3209365982
Wikidata QID: Q112121579
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DOI: 10.1007/S00365-021-09560-3
description: scientific article; zbMATH DE number 7634049