Subword complexity and power avoidance

Abstract

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that –the Thue–Morse word has the minimum possible subword complexity over all overlap-free binary words and all (7/3)-power-free binary words, but not over all (7/3)+-power-free binary words; –the twisted Thue–Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all (7/3)-power-free binary words; –if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; –the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all symmetric square-free ternary words.