Fast Computation of the Arnold Complexity of Length $2^{n}$ Binary Words

Analyzing word complexity usually involves studying the fragments of a word or the process of its construction (see [2] for instance). Arnold introduced [1] a new concept of complexity of a word. The measure of this complexity is determined by the “stability” of a word under the iterated action of a certain operator. Consider an arbitrary nonperiodic binary word w = x1x2 . . . x2n , with w 6= v and k ≥ 2, of length |w| = 2 for n ≥ 1. Denote by (w) the infinite periodic word (w) = ww . . .. Henceforth, by a “word (w)” we understand “an infinite periodic word (w)”. Consider the scheme (word chain)