Language structure of pattern Sturmian words

Pattern Sturmian words introduced by Kamae and Zamboni [Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynamical Systems 22 (2002) 1191–1199; Maximal pattern complexity for discrete systems, Ergodic Theory Dynamical Systems 22 (2002) 1201–1214] are an analogy of Sturmian words for the maximal pattern complexity instead of the block complexity. So far, two kinds of recurrent pattern Sturmian words are known, namely, rotation words and Toeplitz words. But neither a structural characterization nor a reasonable classification of the recurrent pattern Sturmian words is known. In this paper, we introduce a new notion, pattern Sturmian sets, which are used to study the language structure of pattern Sturmian words. We prove that there are exactly two primitive structures for pattern Sturmian words. Consequently, we suggest a classification of pattern Sturmian words according to structures of pattern Sturmian sets and prove that there are at most three classes in this classification. Rotation words and Toeplitz words fall into two different classes, but no examples of words from the third class are known.