Connectedness of fractals associated with Arnoux–Rauzy substitutions
1 LIAFA, CNRS, Université Paris
Diderot, Case 7014, 75205
Paris Cedex 13,
2 Department of Mathematics, University of Turku 20014, Finland.
3 INRIA, Centre Rennes-Bretagne Atlantique, Dyliss, Rennes, France.
Accepted: 14 March 2014
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.
Mathematics Subject Classification: 68R15 / 37B10
Key words: Rauzy fractals / Arnoux–Rauzy substitutions / discrete planes
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