Recognizability of substitutions and complexity of automatic sequences

We first recall some notions and one theorem of recognizability about infinite words fixed points of primitive substitutions. When the length of the substitution is constant equal to q, we study the complexity function p(n) of the fixed point u, which is the number of factors of u of length n. We give a method to compute p(n) with linear recurrence formulas and we prove that the sequence (p(n + 1) - p(n))(n is an element of N) is q-automatic.