Words containing all permutations of a family of factors

TL;DR

This paper proves that infinite words containing all permutations of a family of factors are either periodic or have superlinear complexity, extending previous results on the structure of such words.

Contribution

It generalizes a lemma by de Luca and Zamboni, showing that words containing all permutations of a family of factors are either periodic or have complexity growth faster than linear.

Findings

Infinite words with all permutations of a factor family are either periodic or have superlinear complexity.

Generalizes previous results on Sturmian words and factor permutations.

Provides a broader understanding of the structure of recurrent infinite words.

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.