Subword complexities and iteration

This paper is an overview of results' on the subword complexity of various classes of languages obtained by iterating a mapping, and more precisely on the asymptotic behaviour of this complexity. In the first section we consider iterated morphisms, that is DOL languages. In this area the subword complexity falls into one of five classes, and class membership can be determined. The second section is devoted to homomorphic images 9f DOL languages, that is HDoL languages. In this case there is an infinite hierarchy of complexity classes, and many questions remain unsolved; In section 3 we are concerned with the effect of various combinatorial restrictions (square-freeness, constant distribution} on the subword complexity. Finally in section 4 we give some results concerning iterated sequential mappings(d.g.8.m.); Most proofs are omitted if they appear somewhere else, or only sketched. The reader is assumed to be familiar with basic DOL language theory (see [RSJ). 1. Subword complexity and iterated morphisms The subword complexity (or complexity for short) of a language L is the function h where h (n) is the number of distinct subwords of L with length n. An infinite word S is a mapping from the non-negative integers to some finite alphabet. One can define the complexity of S by the complexity of the language of its prefixes. A DOL system is a 3-tuple D =< X, g, a > where X is a finite alphabet, g: X·'-"+ X·" amorphisill and aE X+ the axiom. The DOL language generated" by Dis L = {l(a), i > O}.