Syndeticity and independent substitutions

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence x is generated by two independent substitutions — at least one being of exponential growth. Then the factors of x appearing infinitely often in x appear with bounded gaps. As an application, we derive an analogue of Cobham’s theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential.