## Asymptotic behavior of growth functions of D0L-systems

##### Date

##### Authors

Cassaigne, Julien

Mauduit, Christian

Nicolas, Francois

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

##### Description

A D0L-system is a triple (A, f, w) where A is a finite alphabet, f is an
endomorphism of the free monoid over A, and w is a word over A. The
D0L-sequence generated by (A, f, w) is the sequence of words (w, f(w), f(f(w)),
f(f(f(w))), ...). The corresponding sequence of lengths, that is the function
mapping each non-negative integer n to |f^n(w)|, is called the growth function
of (A, f, w). In 1978, Salomaa and Soittola deduced the following result from
their thorough study of the theory of rational power series: if the
D0L-sequence generated by (A, f, w) is not eventually the empty word then there
exist a non-negative integer d and a real number b greater than or equal to one
such that |f^n(w)| behaves like n^d b^n as n tends to infinity. The aim of the
present paper is to present a short, direct, elementary proof of this theorem.

Comment: Might appear in the book "Combinatorics, Automata and Number Theory", which is in preparation

Comment: Might appear in the book "Combinatorics, Automata and Number Theory", which is in preparation

##### Keywords

Computer Science - Discrete Mathematics