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On cut sets of attractors of iterated function systems


Authors: Benoît Loridant, Jun Luo, Tarek Sellami and Jörg M. Thuswaldner
Journal: Proc. Amer. Math. Soc. 144 (2016), 4341-4356
MSC (2010): Primary 28A80, 52C20, 54D05
DOI: https://doi.org/10.1090/proc/13182
Published electronically: May 31, 2016
MathSciNet review: 3531184
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Abstract: In this paper, we study cut sets of attractors of iterated function systems (IFS) in $ \mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature.


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Additional Information

Benoît Loridant
Affiliation: Montanuniversität Leoben, Franz Josef Strasse 18, Leoben 8700, Austria

Jun Luo
Affiliation: Department of Statistics, Sun Yat-Sen University, Guangzhou 512075, People’s Republic of China

Tarek Sellami
Affiliation: Department of mathematics, Faculty of sciences of Sfax, Sfax University, Route Soukra, BP 802, 3018 Sfax, Tunisia

Jörg M. Thuswaldner
Affiliation: Montanuniversität Leoben, Franz Josef Strasse 18, Leoben 8700, Austria

DOI: https://doi.org/10.1090/proc/13182
Received by editor(s): December 4, 2014
Received by editor(s) in revised form: December 7, 2015
Published electronically: May 31, 2016
Additional Notes: The first and fourth authors were supported by the project P22855 of the FWF (Austrian Science Fund), the project I 1136 of the FWF and the ANR (French National Research Agency). The second author was supported by the projects 10971233 and 11171123 of the Chinese National Natural Science Foundation.
Communicated by: Michael Wolf
Article copyright: © Copyright 2016 American Mathematical Society