Asymptotic behavior of growth functions of D0L-systems

TL;DR

This paper provides a simple, elementary proof of a theorem describing the asymptotic growth behavior of D0L-systems, showing that their length sequences grow like a polynomial times an exponential.

Contribution

It offers a concise, direct proof of the known asymptotic growth characterization of D0L-systems, simplifying previous complex proofs.

Findings

Growth functions behave like n^d b^n for large n.

Existence of integers d and real b ≥ 1 describing the growth.

The proof is elementary and more accessible.

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.