Counterexamples for IFS-attractors

Nowak, Magdalena; Fernández–Martínez, M.

doi:10.1016/j.chaos.2015.12.006

Cited by 12 publications

(4 citation statements)

References 15 publications

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“…However there are known examples of attractors for wIFSs, which are not attractors for IFSs (see [6]), in our considerations we will need one in wL d Proof. By Lemma 2.2 it suffices to find…”

Section: Resultsmentioning

confidence: 99%

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Comparison of the sets of attractors for systems of contractions and weak contractions

Klinga,

Kwela

2021

Preprint

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For n, d ∈ N we consider the families:• L d n of attractors for iterated function systems (IFS) consisting of n contractions acting on [0, 1] d ,• wL d n of attractors for weak iterated function systems (wIFS) consisting of n weak contractions acting on [0, 1] d .We study closures of the above families as subsets of the hyperspace K([0, 1] d ) of all compact subsets of [0, 1] d equipped in the Hausdorff metric. In particular, we show thatWhat is more, we construct a compact set belonging to L d 2 which is not an attractor for any wIFS. We present a diagram summarizing our considerations.

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Section: Resultsmentioning

confidence: 99%

“…Obviously, each attractor of some IFS is an attractor of some wIFS. The reverse inclusion is not true by [6].…”

Section: Preliminariesmentioning

confidence: 93%

Comparison of the sets of attractors for systems of contractions and weak contractions

Klinga,

Kwela

2021

Preprint

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“…Clearly, an IFS attractor is a weak IFS attractor. However, the reverse inclusion is not true by [11].…”

Section: S(a)mentioning

confidence: 95%

Porosities of the sets of attractors

Klinga¹,

Kwela²

2021

Preprint

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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 A w [0, 1] of attractors for weak iterated function systems acting on [0, 1].It is known that both A[0, 1] and A w [0, 1] are meager subsets of the hyperspace K([0, 1]) (of all compact subsets of [0, 1] equipped in the Hausdorff metric). Actually, A[0, 1] is even σ-lower porous while the question about σ-lower porosity of A w [0, 1] is still open.We prove that A[0, 1] is not σ-strongly porous in K([0, 1]). Moreover, we show that A w [0, 1] \ A[0, 1] is dense in K([0, 1]).

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“…Obviously, each attractor of some IFS is an attractor of some wIFS. However, the reverse inclusion is not true -see [7] for counterexamples.…”

Section: Weak Ifs Attractorsmentioning

confidence: 99%

Borel complexity of the family of attractors for weak IFSs

Klinga¹,

Kwela²

2021

Preprint

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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 in the Hausdorff metric). We prove that wIFS d is G δσ -hard in K([0, 1] d ), for all d ∈ N. In particular, wIFS d is not F σδ (in contrast to the family IFS d of attractors for classical iterated function systems acting on [0, 1] d , which is F σ ). Moreover, we show that in the one-dimensional case, wIFS 1 is an analytic subset of K([0, 1]).

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Counterexamples for IFS-attractors

Cited by 12 publications

References 15 publications

Comparison of the sets of attractors for systems of contractions and weak contractions

Comparison of the sets of attractors for systems of contractions and weak contractions

Porosities of the sets of attractors

Borel complexity of the family of attractors for weak IFSs

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