Abstract
This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS\(^d\) of attractors for weak iterated function systems acting on \([0,1]^d\) (as a subset of the hyperspace \(K([0,1]^d)\) of all compact subsets of \([0,1]^d\) equipped with the Hausdorff metric). We prove that wIFS\(^d\) is \(G_{\delta\sigma}\)-hard in \(K([0,1]^d)\), for all \({d\in\mathbb{N}}\). In particular,wIFS\(^d\) is not \(F_{\sigma\delta}\) (in contrast to the family IFS\(^d\) of attractors for classical iterated function systems acting on \([0,1]^d\), which is \(F_{\sigma}\)). Moreover, we show that in the one-dimensional case, wIFS\(^1\) is an analytic subset of \(K([0,1])\).
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References
E. D’Aniello and T. H. Steele, Attractors for iterated function schemes on \([0,1]^N\) are exceptional, J. Math. Anal. Appl., 424 (2015), 534–541.
E. D’Aniello and T. H. Steele, Attractors for iterated function systems, J. Fractal Geom., 3 (2016), 95–117.
E. D’Aniello and T. H. Steele, Attractors for classes of iterated function systems, European J. Math., 5 (2019), 116–137.
M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 1 (1962), 74–79.
A. S. Kechris, Classical Descriptive Set Theory, Springer (New York, 1998).
P. Klinga, A. Kwela and M. Staniszewski, Size of the set of attractors for iterated function systems, Chaos Solitons Fractals, 128 (2019), 104–107.
M. Nowak and M. Fernández-Martínez, Counterexamples for IFS-attractors, Chaos Solitons Fractals, 89 (2016), 316–321.
S. Solecki, Analytic ideals, Bull. Symbolic Logic, 3 (1996), 339–348.
S. M. Srivastava, A Course on Borel Sets, Springer (New York, 1998).
L. L. Stachó and L. I. Szabó, A note on invariant sets of iterated function systems, Acta Math. Hungar., 119 (2008), 159–164.
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Klinga, P., Kwela, A. Borel complexity of the family of attractors for weak IFSs. Acta Math. Hungar. 166, 124–141 (2022). https://doi.org/10.1007/s10474-021-01199-7
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DOI: https://doi.org/10.1007/s10474-021-01199-7