Balance and Abelian complexity of the Tribonacci word

TL;DR

This paper investigates the Abelian complexity of the Tribonacci word, establishing its possible values and proving its 2-balance property through combinatorial and spectral methods.

Contribution

It provides the first published proof of the Tribonacci word's 2-balance property and characterizes its Abelian complexity values.

Findings

AC(n) takes values in {3,4,5,6,7} for all n

Each value in {3,4,5,6,7} occurs at least once

The Tribonacci word is 2-balanced

Abstract

G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $R^2$, each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity AC(n) of the Tribonacci word $t$ which is the unique fixed point of $\tau$. We show that $AC(n)\in {3,4,5,6,7}$ for each $n\geq 1$, and that each of these five values is assumed. Our proof relies on the fact that the Tribonacci word is 2-balanced, i.e., for all factors $U$ and $V$ of $t$ of equal length, and for every letter $a \in {0,1,2}$, the number of occurrences of $a$ in $U$ and the number of occurrences of $a$ in $V$ differ by at most 2. While this result is announced in several papers, to the best of our knowledge no proof of this fact has ever been published. We offer two very different proofs of the 2-balance property of $t$. The first uses the word combinatorial properties of the generating morphism, while the second exploits the spectral properties of the incidence matrix of $\tau$.