On maximal pattern complexity of some automatic words

TETURO KAMAE, PAVEL V. SALIMOV
2010 Ergodic Theory and Dynamical Systems  
The pattern complexity of a word for a given pattern S, where S is a finite subset of {0, 1, 2, . . .}, is the number of distinct restrictions of the word to S + n (with n = 0, 1, 2, . . .). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys. 22 (4) (2002) , 1191-1199], is the maximum value of the pattern complexity of S with #S = k as a function of k = 1,
more » ... , . . . . A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0, 1} (as a function of k = 1, 2, . . .) is either bounded, a linear function of k, or 2 k . Proof. Let x = ψ ω (0) be such a point. We shall construct by induction a series of windows T n such that F x (T n ) ⊃ 0 n−1 .
doi:10.1017/s0143385710000453 fatcat:rgpbmlbrwzfnvjgejaro2vprzi