Words containing all permutations of a family of factors

Abstract

We prove that if a uniformly recurrent infinite word contains as a factor any finite permutation of words from an infinite family, then either this word is periodic, or its complexity (that is, the number of factors) grows faster than linearly. This result generalizes one of the lemmas of a recent paper by de Luca and Zamboni, where it was proved that such an infinite word cannot be Sturmian.