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Connectedness of fractals associated with Arnoux–Rauzysubstitutions

Published online by Cambridge University Press:  27 May 2014

Valérie Berthé ,
Timo Jolivet  and
Anne Siegel
Valérie Berthé
Affiliation:
LIAFA, CNRS, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France.. berthe@liafa.univ-paris-diderot.fr
Timo Jolivet
Affiliation:
LIAFA, CNRS, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France.. berthe@liafa.univ-paris-diderot.fr Department of Mathematics, University of Turku 20014, Finland.
Anne Siegel
Affiliation:
INRIA, Centre Rennes-Bretagne Atlantique, Dyliss, Rennes, France.
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Abstract

Rauzy fractals are compact sets with fractal boundary that can be associated with anyunimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorfflimit of a sequence of compact sets, where each set is a renormalized projection of afinite union of faces of unit cubes. We exploit this combinatorial definition to prove theconnectedness of the Rauzy fractal associated with any finite product of three-letterArnoux–Rauzy substitutions.

Keywords

Rauzy fractalsArnoux–Rauzy substitutionsdiscrete planes

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