Ancestors graph and an upper bound for the subword complexity function

A general upper bound for the subword complexity function of a fixed point of a primitive substitution is presented. The approach relies on the introduction and discussion of the ancestors graph of a substitution. More precisely, we characterize the language of a fixed point of a primitive substitution as the strongly connected component of the letters in its ancestors graph; we propose algorithms for the local construction of this graph; we give an upper bound for the subword complexity function in terms of the Hilbert series of some quotient of a noncommutative polynomial ring by a sum of automatic bilateral ideals.