Asymptotic properties of free monoid morphisms

TL;DR

This paper investigates the asymptotic behavior of morphic words with erasing morphisms, extending Cobham's results by analyzing growth types of iterated morphisms and providing algorithms to compute related morphisms.

Contribution

It offers a detailed study of growth types of morphic words with non-erasing morphisms and presents an explicit algorithm for computing associated morphisms from given data.

Findings

Characterization of growth types of iterated morphisms

Extension of Cobham's theorem to erasing morphisms

Algorithm for computing morphisms $\sigma$ and $ au$

Abstract

Motivated by applications in the theory of numeration systems and recognizable sets of integers, this paper deals with morphic words when erasing morphisms are taken into account. Cobham showed that if an infinite word $w =g(f^\omega(a))$ is the image of a fixed point of a morphism $f$ under another morphism $g$, then there exist a non-erasing morphism $\sigma$ and a coding $\tau$ such that $w =\tau(\sigma^\omega(b))$. Based on the Perron theorem about asymptotic properties of powers of non-negative matrices, our main contribution is an in-depth study of the growth type of iterated morphisms when one replaces erasing morphisms with non-erasing ones. We also explicitly provide an algorithm computing $\sigma$ and $\tau$ from $f$ and $g$.