We study the topological properties of attractors of iterated function systems (IFS) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation IFS: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, IFS (of first generation), or it possesses a particular dissection property, the attractor of the second generation IFS is the union of a finite number of closed intervals. We also prove a theorem that generalizes this result to certain in finite sums of compact sets, in the sense of Minkowski and under the Hausdorff metric.

Mantica, G., Peirone, R. (2017). Attractors of iterated function systems with uncountably many maps and infinite sums of Cantor sets. JOURNAL OF FRACTAL GEOMETRY, 4(3), 215-256 [10.4171/JFG/50].

Attractors of iterated function systems with uncountably many maps and infinite sums of Cantor sets

Peirone, Roberto
2017-01-01

Abstract

We study the topological properties of attractors of iterated function systems (IFS) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation IFS: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, IFS (of first generation), or it possesses a particular dissection property, the attractor of the second generation IFS is the union of a finite number of closed intervals. We also prove a theorem that generalizes this result to certain in finite sums of compact sets, in the sense of Minkowski and under the Hausdorff metric.
2017
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Cantor sets; iterated function systems; second generation IFS; attractors; Minkowski sums; series of Cantor sets
Mantica, G., Peirone, R. (2017). Attractors of iterated function systems with uncountably many maps and infinite sums of Cantor sets. JOURNAL OF FRACTAL GEOMETRY, 4(3), 215-256 [10.4171/JFG/50].
Mantica, G; Peirone, R
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/245575
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