On prefixal factorizations of words

We consider the class P 1 of all infinite words x ? A ω over a finite alphabet A admitting a prefixal factorization, i.e., a factorization x = U 0 U 1 U 2 ? where each U i is a non-empty prefix of x . With each x ? P 1 one naturally associates a "derived" infinite word ? ( x ) which may or may not admit a prefixal factorization. We are interested in the class P ∞ of all words x of P 1 such that ? n ( x ) ? P 1 for all n ? 1 . Our primary motivation for studying the class P ∞ stems from its connection to a coloring problem on infinite words independently posed by T. Brown and by the second author. More precisely, let P be the class of all words x ? A ω such that for every finite coloring ? : A + ? C there exist c ? C and a factorization x = V 0 V 1 V 2 ? with ? ( V i ) = c for each i ? 0 . In a recent paper (de Luca et?al., 2014), we conjectured that a word x ? P if and only if x is purely periodic. In this paper we prove that P ? P ∞ , so in other words, potential candidates to a counter-example to our conjecture are amongst the non-periodic elements of P ∞ . We establish several results on the class P ∞ . In particular, we prove that a Sturmian word x belongs to P ∞ if and only if x is nonsingular, i.e., no proper suffix of x is a standard Sturmian word.