Coloring Factors of Substitutive Infinite Words

In this paper, we consider infinite words that arise as fixed points of primitive substitutions on a finite alphabet and finite colorings of their factors. Any such infinite word exhibits a "hierarchal structure" that will allow us to define, under the additional condition of strong recognizability, certain remarkable finite colorings of its factors. In particular, we generalize two combinatorial results by Justin and Pirillo concerning arbitrarily large monochromatic $k$-powers occurring in infinite words, in view of a recent paper by de Luca, Pribavkina and Zamboni, we will give new examples of classes of infinite words $\mathbf{u}$ and finite colorings that do not allow infinite monochromatic factorizations $\mathbf{u}=\mathbf{u}_1\mathbf{u}_2\mathbf{u}_3\ldots$.