Cyclic Complexity of Some Infinite Words and Generalizations

Abstract

Cassaigne et al. introduced the cyclic complexity function c_x(n), which gives the number of cyclic conjugacy classes of length-n factors of a word x. We study the behavior of this function for the Fibonacci word f and the Thue–Morse word t. If φ = (1 + √5)/2, we show that lim sup_{n → 1} c_f(n)/n ≥ 2/φ² and conjecture that equality holds. Similarly, we show that lim sup_{n → 1} c_t(n)/n ≥ 2 and conjecture that equality holds. We also propose a generalization of the cyclic complexity function and suggest some directions for further investigation. Most results are obtained by computer proofs using Mousavi’s Walnut software.