Abstract

We prove that, given a double sequence w over the alphabet A (i.e. a mapping from Z Z 2 to A), if there exists a pair (n 0 ; m 0 ) 2 Z Z 2 such that p w (n 0 ; m 0 ) ! 1 100 n 0 m 0 , then w has a periodicity vector, where p w of w is the complexity function of w. 1 Introduction In combinatorics on words the notions of complexity and periodicity are of fundamental importance. The complexity function of a formal language counts, for any natural number n, the number of words in the language of length n. The complexity function of a word (finite, infinite, biinfinite) is the complexity function of the formal language whose elements are all the factors (or blocks, or also subwords) of the word. The Morse-Hedlund Theorem states that there exists an important relationship between periodicity and complexity. In particular it states that for any biinfinite word w if the number of its different factors of length n is less Dipartimento di Matematica ed Applicazioni, Universit`a degl...

Citations