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.The first author was supported by an NSERC USRA. The second author was supported by an NSERC Discovery Grant