Abelian complexity and Abelian co-decomposition

TL;DR

This paper introduces a new technique for analyzing the abelian complexity of certain infinite words, confirming that the Tribonacci word's abelian complexity takes specific values infinitely often.

Contribution

It presents a novel method for studying abelian complexity and resolves an open question about the Tribonacci word's complexity values.

Findings

The abelian complexity of the Tribonacci word attains values 4, 5, and 6 infinitely often.

The proposed technique effectively explores abelian complexity in words related to Parry numbers.

The open question by Richomme, Saari, and Zamboni is affirmatively answered.

Abstract

We propose a technique for exploring the abelian complexity of recurrent infinite words, focusing particularly on infinite words associated with Parry numbers. Using that technique, we give the affirmative answer to the open question posed by Richomme, Saari and Zamboni, whether the abelian complexity of the Tribonacci word attains each value in $\{4,5,6\}$ infinitely many times.