THREE COMPLEXITY FUNCTIONS

Abstract

International audienceFor an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes. Given a language L on a finite alphabet A, the complexity function p L (n) counts for every n the number of factors of length n of L; this is a very useful notion, both inside word combinatorics and for the study of symbolic dynamical systems, see for example the survey [7]; of particular interest are the infinite words which are determined by the complexity of their language, those words for which p L (n) ≤ n for at least one n are ultimately periodic [15], while the Sturmian words, of complexity n + 1 for all n, are natural codings of rotations, see [6, 16], or Chapter 6 of [17], and Section 4 below. Note that the complexity is exponential when the language has positive topological entropy, and has not been widely used for that range of languages. To study further the combinatorial properties of infinite words, the notion of maximal pattern complexity, denoted by p