Factor complexity of infinite words associated with non-simple Parry numbers

TL;DR

This paper investigates the factor complexity of infinite words linked to non-simple Parry numbers, introducing methods to analyze special factors and characterizing when these words exhibit affine complexity.

Contribution

It provides a new method for determining infinite left special branches and characterizes non-simple Parry numbers with affine complexity in their associated infinite words.

Findings

Method for determining infinite left special branches

Characterization of non-simple Parry numbers with affine complexity

Introduction of $(a,b)$-maximal left special factors

Abstract

The factor complexity of the infinite word $\ubeta$ canonically associated to a non-simple Parry number $\beta$ is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words $\ubeta$ only. To complete the description of its special factors, we define and study $(a,b)$-maximal left special factors. This enables us to characterize non-simple Parry numbers $\beta$ for which the word $\ubeta$ has affine complexity.