Abelian maximal pattern complexity of words

Abstract

In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words

Abelian Nivat's conjecture for non-rectangular patterns

“…So, among aperiodic words, Sturmian words have minimal complexity both in classical and in abelian sense. The study of abelian complexity of one-dimensional infinite words has been developed, e.g., in [10,15,23,24]. For two-dimensional words, the abelian modifications of Nivat's conjecture has been studied in [19].…”

“…The maximal pattern complexity p * w of a word w is defined as a function counting, for each k, the supremum of the pattern complexities for patterns defined by figures of size k. Similarly to factor complexity and abelian complexity, the maximal pattern complexity also gives a characterization of periodicy in the one-dimensional case: An infinite one-dimensional word w is eventually periodic if and only if p * w (k) < 2k for some integer k [11]. The abelian maximal pattern complexity also gives a characterization of aperiodicty in terms of so-called aperiodicity by projection [9,10].…”