Porosities of the sets of attractors

This paper is another attempt to measure the difference between the family A[0,1] of attractors for iterated function systems acting on [0,1] and a broader family, the set Aw[0,1] of attractors for weak iterated function systems acting on [0,1]. It is known that both A[0,1] and Aw[0,1] are meager su...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2022-10, Vol.514 (2), p.126348, Article 126348
Main Authors: Klinga, Paweł, Kwela, Adam
Format: Article
Language:English
Subjects:
Attractors
Banach fractals
Iterated function systems
σ-lower porosity
σ-strong porosity
σ-upper porosity
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Summary:This paper is another attempt to measure the difference between the family A[0,1] of attractors for iterated function systems acting on [0,1] and a broader family, the set Aw[0,1] of attractors for weak iterated function systems acting on [0,1]. It is known that both A[0,1] and Aw[0,1] are meager subsets of the hyperspace K([0,1]) (of all compact subsets of [0,1] equipped with the Hausdorff metric). Moreover, A[0,1] is σ-lower porous while the question about σ-lower porosity of Aw[0,1] is still open. We prove that A[0,1] is not σ-strongly porous in K([0,1]). Moreover, we show that Aw[0,1]∖A[0,1] is dense in K([0,1]).
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126348