On a class of infinite words with affine factor complexity

In this article, we consider the fixed point of a primitive substitution canonically defined by a β-numeration system. The problem of determination of the factor complexity of such an infinite word has been solved only partially. Here we provide a necessary and sufficient condition on the Rényi expansion of one for having an affine factor complexity map C(n), that is, such that C(n)=an+b for any positive integer n.