Iterated function systems enriched with symmetry
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.
scientific article; zbMATH DE number 7634049
scientific article