Asymptotic behavior of growth functions of D0L-systems
Abstract
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.
Cite
@article{arxiv.0804.1327,
title = {Asymptotic behavior of growth functions of D0L-systems},
author = {Julien Cassaigne and Christian Mauduit and Francois Nicolas},
journal= {arXiv preprint arXiv:0804.1327},
year = {2009}
}
Comments
Might appear in the book "Combinatorics, Automata and Number Theory", which is in preparation