On the existence of the Hutchinson measure for generalized iterated function systems
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On the Existence of the Hutchinson Measure for Generalized Iterated Function Systems Filip Strobin1 Received: 15 May 2020 / Accepted: 23 August 2020 © The Author(s) 2020
Abstract We prove that each generalized (in the sense of Miculescu and Mihail) IFS consisting of contractive maps generates the unique generalized Hutchinson measure. This result extends the earlier result due to Miculescu and Mihail in which the assertion is proved under certain additional contractive assumptions. Keywords Generalized iterated function systems · Hutchinson measure · Markov operator · Monge–Kantorovich metric Mathematics Subject Classification Primary: 28A80; Secondary: 47H09 · 28A33
1 Introduction In the last decade, various aspects of the theory of classical iterated iterated function systems (IFSs) has been extended to the framework of generalized IFSs (GIFSs for short), which were introduced in 2008 by Miculescu and Mihail (see [13,15,16]). Instead of selfmaps of a given metric space X , GIFSs consist of maps defined on the finite Cartesian product X m with values in X . It turned out that many classical results for IFSs have natural counterparts in the GIFSs setting (see, e.g., [17,23,24]). In particular, GIFSs consisting of contractive maps generate unique compact sets which can be called as their attractors. On the other hand, the class of GIFSs’ attractors is essentially wider than the class of IFSs’ attractors (see [10,22]). One of a very important parts of the IFS theory bases on the existence of the socalled Hutchinson measure, which is the unique measure invariant w.r.t. the so-called Markov operator generated by the underlying IFS (see, e.g., [5]). Miculescu and Mihail in [12,14] proved that a GIFS F generates a counterpart of the Hutchinson measure, provided that the underlying maps from F satisfy some
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Filip Strobin [email protected] Institute of Mathematics, Lodz University of Technology, Wólcza´nska 215, 93-005 Lodz, Poland 0123456789().: V,-vol
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additional contractive assumptions (a bit different approach to the generalized IFSs and their Hutchinson measures were considered in [19]). Using a completely different approach, in the main result of our paper we will prove the same assertion but without this additional requirements. We also give an example which shows that the original proof from [14] cannot work in the general case. Presented reasonings are inspired by the ones from [8] and they use the machinery of the code spaces for GIFSs from [24].
2 Preliminaries 2.1 Generalized IFSs and a counterpart of the Hutchinson–Barnsley theorem Assume that (X , d) is a metric space and m ∈ N. Consider the Cartesian product X m as a metric space with the maximum metric dm . By K(X ) we will denote the hyperspace of all nonempty and compact subsets of X , endowed with the Hausdorff–Pompeiu metric h. Definition 2.1 By a generalized iterated function system of order m (GIFS for short) we will mean any finite family F of continuous maps defined on X m and with values in X . Each GIFS
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