On the existence of the Hutchinson measure for generalized iterated function systems

Strobin, Filip

doi:10.1007/s12346-020-00420-2

Cited by 7 publications

(2 citation statements)

References 18 publications

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“…It is worth to mention that Theorem 3.9 was fairly improved by [Str20], Theorem 4.3, proving that S admits the Hutchinson measure µ S and supp(µ S ) = A S under the hypothesis that each map of the GIFS is a generalized Matkowski contraction, using new techniques and code spaces. Definition 3.10.…”

Section: Hutchinson (Invariant) Measures Of Gifsmentioning

confidence: 99%

The Hutchinson-Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

Oliveira¹

2022

Preprint

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The study of generalized iterated function systems (GIFS) was introduced by Mihail and Miculescu in 2008. We provide a new approach to study those systems as the limit of the Hutchinson-Barnsley theory for infinite iterated function systems (IIFS) which was developed by many authors in the last years. We show that any attractor of a contractive generalized iterated function system is the limit w.r.t Hausdorff-Pompeiu metric of attractors of contractive infinite iterated function systems. We also prove that any Hutchinson measure for a contractive generalized iterated function system with probabilities is the limit w.r.t. the Monge-Kantorovich metric of the Hutchinson measures for contractive infinite iterated function systems with probabilities.

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Section: Hutchinson (Invariant) Measures Of Gifsmentioning

confidence: 99%

The Hutchinson-Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

Oliveira¹

2022

Preprint

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“…For more details regarding Monge-Kantorovich norm, one can consult [6][7][8][9][10][11][12][13]. About the fractals theory, you can read the following [14][15][16][17]. For more details regarding functional analysis, see [16,18,19].…”

Section: Introductionmentioning

confidence: 99%

About Some Monge–Kantovorich Type Norm and Their Applications to the Theory of Fractals

Mierlus-Mazilu

Niţă

2022

Mathematics

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If X is a Hilbert space, one can consider the space cabv(X) of X valued measures defined on the Borel sets of a compact metric space, having a bounded variation. On this vector measures space was already introduced a Monge–Kantorovich type norm. Our first goal was to introduce a Monge–Kantorovich type norm on cabv(X), where X is a Banach space, but not necessarily a Hilbert space. Thus, we introduced here the Monge–Kantorovich type norm on cabvLq([0,1]),(1<q<∞). We obtained some properties of this norm and provided some examples. Then, we used the Monge–Kantorovich norm on cabvKn(K being R or C) to obtain convergence properties for sequences of fractal sets and fractal vector measures associated to a sequence of iterated function systems.

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The Invariant Measure for a Countable Generalized Iterated Function System

Abraham

2024

Mediterr. J. Math.

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The aim of this paper is to answer one of the open questions raised in Strobin [Qual. Theory Dyn. Syst. 19, 85 (2020)] of whether there exists an invariant (Hutchinson) measure for generalized iterated function systems of any order, consisting of a countably infinite number of maps. Our results likewise strengthen those obtained in Secelean [Mediterr. J. Math. 11, 361–372 (2014)], where the existence of the invariant measure is ascertained only for the case of generalized iterated function systems of order 2, consisting of functions which satisfy a particular contractive condition.

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On the existence of the Hutchinson measure for generalized iterated function systems

Cited by 7 publications

References 18 publications

The Hutchinson-Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

The Hutchinson-Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

About Some Monge–Kantovorich Type Norm and Their Applications to the Theory of Fractals

The Invariant Measure for a Countable Generalized Iterated Function System

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