Combinatorial and Computational aspects of Tilings

When we consider Pisot primitive unimodular irreducible substitution over two letters, we know connectedness of its atomic surface thanks to the following theorem:

Theorem 1. [1] Let σ be a primitive unimodular substitution over {1, 2}. Then the Rauzy fractals X1, X2 and X1 ∪ X2 are intervals if and only if σ is invertible.

But we don’t know how invertibility works on the property of connectedness in the case of three letters or in the case of reducible case. I will discuss these cases showing some examples.