  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.
