Connectedness of fractals associated with Arnoux-Rauzy substitutions (Q2874636)

It is know from the work of \textit{P. Arnoux} and \textit{S. Ito} [Bull. Belg. Math. Soc. - Simon Stevin 8, No. 2, 181--207 (2001; Zbl 1007.37001)] that for every Pisot irreducible substitution on a finite set of letters one can associate a Rauzy fractal, that is, a certain compact set having fractal boundary. In the present paper the authors prove that the Rauzy fractal associated with finite products of three-letter Arnoux-Rauzy substitutions is a connected set. The proof is based on a combinatorial characterization of Rauzy fractals due to Arnoux and Ito [loc. cit.]. The paper ends with a section on further questions and the presentation of some counterexamples.