Connectedness of fractals associated with Arnoux-Rauzy substitutions
Val\'erie Berth\'e, Timo Jolivet, Anne Siegel
TL;DR
This paper proves that Rauzy fractals linked to finite products of three-letter Arnoux-Rauzy substitutions are connected, using a combinatorial approach based on their Hausdorff limit construction.
Contribution
It establishes the connectedness of these Rauzy fractals for the first time through a novel combinatorial proof.
Findings
Connectedness of Rauzy fractals for Arnoux-Rauzy substitutions
Hausdorff limit approach to fractal boundary analysis
New combinatorial proof technique
Abstract
Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux-Rauzy substitutions.
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