[Substitution topologique pour la pavage fractal apériodique de Rauzy]
We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature, defined in [6]). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.
On considère deux familles de pavages auto-similaires de nature différente : ceux obtenus par translation de copies d’un ensemble fractal défini par un système de fonctions itérées, et ceux obtenus comme la réalisation géométrique d’une substitution topologique (un objet purement combinatoire, défini dans [6]). On établit un lien entre les deux familles dans un cas particulier, en définissant une substitution topologique explicitement puis en démontrant qu’elle engendre les mêmes pavages que ceux associés au fractal Tribonacci de Rauzy.
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Keywords: Rauzy fractal, tiling, tribonacci fractal, topological substitution, combinatorial substitution.
Mots-clés : Rauzy fractal, tiling, fractal tribonacci, substitution topologique, substitution combinatoire
Bédaride, Nicolas 1 ; Hilion, Arnaud 1 ; Jolivet, Timo 2
@article{BSMF_2018__146_3_575_0,
author = {B\'edaride, Nicolas and Hilion, Arnaud and Jolivet, Timo},
title = {Topological substitution for the aperiodic {Rauzy} fractal tiling},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {575--612},
year = {2018},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {146},
number = {3},
doi = {10.24033/bsmf.2762},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2762/}
}
TY - JOUR AU - Bédaride, Nicolas AU - Hilion, Arnaud AU - Jolivet, Timo TI - Topological substitution for the aperiodic Rauzy fractal tiling JO - Bulletin de la Société Mathématique de France PY - 2018 SP - 575 EP - 612 VL - 146 IS - 3 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2762/ DO - 10.24033/bsmf.2762 LA - en ID - BSMF_2018__146_3_575_0 ER -
%0 Journal Article %A Bédaride, Nicolas %A Hilion, Arnaud %A Jolivet, Timo %T Topological substitution for the aperiodic Rauzy fractal tiling %J Bulletin de la Société Mathématique de France %D 2018 %P 575-612 %V 146 %N 3 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2762/ %R 10.24033/bsmf.2762 %G en %F BSMF_2018__146_3_575_0
Bédaride, Nicolas; Hilion, Arnaud; Jolivet, Timo. Topological substitution for the aperiodic Rauzy fractal tiling. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 3, pp. 575-612. doi: 10.24033/bsmf.2762
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