Maximal pattern complexity of two-dimensional words

The maximal pattern complexity of one-dimensional words has been studied in several papers [T. Kamae, L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynam. Systems 22(4) (2002) 1191–1199; T. Kamae, L. Zamboni,Maximal pattern complexity for discrete systems, Ergodic Theory Dynam. Systems 22(4) (2002) 1201–1214; T. Kamae, H. Rao, Pattern Complexity over letters, E. Comb. J., to appear; T. Kamae, Y.M. Xue, Two dimensional word with 2k maximal pattern complexity, Osaka J. Math. 41(2) (2004) 257–265]. We study the maximal pattern complexity p∗ (k) of two-dimensional words . A two-dimensional version of the notion of strong recurrence is introduced. It is shown that if is strongly recurrent, then either is doubly periodic or p∗ (k) 2k (k = 1, 2, . . .). Accordingly, we define a two-dimensional pattern Sturmian word as a strongly recurrent word with p∗ (k)= 2k. Examples of pattern Sturmian words are given. © 2006 Elsevier B.V. All rights reserved.