Generalized iterated function systems, multifunctions and Cantor sets

Klimek, Maciej; Kosek, Marta

doi:10.4064/ap96-1-2

Cited by 23 publications

(18 citation statements)

References 12 publications

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“…We will mention some of these extensions. For infinite iterated function systems it is remarkable the work of Henning Fernau [7], K. Leśniak [12], F. Mendivil in [16], G. Gwóźdź-Lukowska and J. Jackymski in [8], M. Klimek and M. Kosek in [11], A. Mihail and R. Miculescu in [21] and others.…”

Section: Introductionmentioning

confidence: 99%

The Existence of the Attractor of Countable Iterated Function Systems

Secelean

2011

Mediterr. J. Math.

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The theory of iterated function systems (IFS) and of infinite iterated function systems consisting of contraction mappings has been studied in the last decades. Some extensions of the spaces and the contractions concern many authors in fractal theory. In this paper there are described some results in that topic concerning the existence and uniqueness of nonempty compact set which is a set "fixed point" of a countable iterated function system (CIFS). Moreover, some approximations of the attractor of a CIFS by the attractors of the partial IFSs are given.

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Section: Introductionmentioning

confidence: 99%

The Existence of the Attractor of Countable Iterated Function Systems

Secelean

2011

Mediterr. J. Math.

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“…This limit is called the attractor of the matrix . By [16,Lemma 4.3] and the remark below it, we have…”

Section: Attractors Of Matrices Of Affine Mappingsmentioning

confidence: 92%

“…It was shown in [16] that the sequence (( 1 • · · · • n )(K )) n∈N of compact sets converges (in the Hausdorff metric) to a compact set E independent of the choice of the set K . This limit is called the attractor of the matrix .…”

Section: Attractors Of Matrices Of Affine Mappingsmentioning

confidence: 99%

How to construct totally disconnected Markov sets?

Białas-Cież

Kosek

2010

Annali di Matematica

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We deduce a polynomial estimate on a compact planar set from a polynomial estimate on its circular projection, which enables us to prove Markov and Bernstein-Walsh type inequalities for certain sets. We construct -totally disconnected Markov sets that are scattered around zero in different directions; -a Markov set E ⊂ R such that neither E ∩ [0, +∞) nor E ∩ (−∞, 0] admit Markov's inequality; -a Markov set that is not uniformly perfect.Finally, we propose a construction based on a generalization of iterated function systems: a way of obtaining a big family of uniformly perfect sets.

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“…-to consider more general domains or ranges of the iterated function systems (see, for example, [6], [10], [13], [14], [24], [30], [33] and [45]). …”

Section: Introductionmentioning

confidence: 99%

Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

Georgescu

2017

Annals of West University of Timisoara - Mathematics and Computer Science

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Abstract. The concept of generalized convex contraction was introduced and studied by V. Istrȃţescu and the notion of b -metric space was introduced by I. A. Bakhtin and S. Czerwik. In this paper we combine these two elements by studying iterated function systems consisting of generalized convex contractions on the framework of b-metric spaces. More precisely we prove the existence and uniqueness of the attractor of such a system providing in this way a generalization of Istrȃţescu's convex contractions fixed point theorem in the setting of complete strong b-metric spaces.

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Generalized iterated function systems, multifunctions and Cantor sets

Cited by 23 publications

References 12 publications

The Existence of the Attractor of Countable Iterated Function Systems

The Existence of the Attractor of Countable Iterated Function Systems

How to construct totally disconnected Markov sets?

Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

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