Profinite approach to S-adic shift spaces II: Invertible directive sequences

This paper is the second in a series about (relatively) free profinite semigroups and S-adic representations of minimal shifts. The first installment introduced the key notions of profinite images and saturating properties. The present paper examines the relationship between relative invertibility conditions (on a directive sequence) and charging properties (of a shift). The main result shows that the two properties are equivalent under mild conditions. Moreover, charging properties are shown to be equivalent to density of subgroups generated by return words in relatively free profinite groups. The main result can then be used to deduce purely discrete statements about subgroups generated by return words in relatively free groups. We also highlight applications of the main result to the study of maximal bifix decoding, following earlier work by the second author. Finally, we establish some results on the effect of direct images on charging properties and explain how to apply these results to the case of episturmian shifts.