Characterization of infinite LSP words and endomorphisms preserving the LSP property

Answering a question of G. Fici, we give an $S$-adic characterization of thefamily of infinite LSP words, that is, the family of infinite words having all their left special factors as prefixes.More precisely we provide a finite set of morphisms $S$ and an automaton ${\cal A}$ such that an infinite word is LSP if and only if it is $S$-adic and one of its directive words is recognizable by ${\cal A}$.Then we characterize the endomorphisms that preserve the property of being LSP for infinite words.This allows us to prove that there exists no set $S'$ of endomorphisms for which the set of infinite LSP words corresponds to the set of $S'$-adic words. This implies that an automaton is required no matter which set of morphisms is used.Comment: arXiv admin note: text overlap with arXiv:1705.0578

On sets of indefinitely desubstitutable words

The stable set associated to a given set S of nonerasing endomorphisms or substitutions is the set of all right infinite words that can be indefinitely desubstituted over S. This notion generalizes the notion of sets of fixed points of morphisms. It is linked to S-adicity and to property preserving morphisms. Two main questions are considered. Which known sets of infinite words are stable sets? Which ones are stable sets of a finite set of substitutions? While bringing answers to the previous questions, some new characterizations of several well-known sets of words such as the set of binary balanced words or the set of episturmian words are presented. A characterization of the set of nonerasing endomorphisms that preserve episturmian words is also provided