On Substitutions Preserving Their Return Sets

Abstract

We consider the question of whether or not a given primitive substitution preserves its sets of return words-or return sets for short. More precisely, we study the property asking that the image of the return set to a word equals the return set to the image of that word. We show that, for bifix encodings (where images of letters form a bifix code), this property holds for all but finitely many words. On the other hand, we also show that every conjugacy class of Sturmian substitutions contains a member for which the property fails infinitely often. Various applications and examples of these results are presented, including a description of the subgroups generated by the return sets in the shift of the Thue-Morse substitution. Up to conjugacy, these subgroups can be sorted into strictly decreasing chains of isomorphic subgroups weaving together a simple pattern. This is in stark contrast with the Sturmian case, and more generally with the dendric case (including in particular the Arnoux-Rauzy case), where it is known that all return sets generate the free group over the underlying alphabet.