On the connectedness of the Rauzy fractal

Abstract

If X is a reducible unimodular Pisot substitution, the Rauzy fractal associated to X can be studied using the strand space. In this dissertation we are going to provide a characterization of the connectedness of the Rauzy Fractal in terms of infinitely many graphs closely related to the proximal structure of the strands in the strand space. Using this characterization, we show a topological characterization of invertible substitutions on two letters, and show that the Rauzy fractal associated to an Arnoux-Rauzy substitution is connected. We show that if two reducible unimodular Pisot substitutions X and Y are homemorphic, then there is a subdivision of the Rauzy fractal for X into finitely many pieces, which, after applying suitable linear transformations and a translations to each piece, becomes a set whose union is the Rauzy fractal for Y. We also found an algorithm to find asymptotic composants.