Balance and Abelian complexity of the Tribonacci word

Abstract

G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau :0\mapsto 01,1\mapsto 02and2\mapsto 0$ 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 $n\ge 1$, 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 $\tau$.

20 pages, 1 figure. This is an extended version of 0904.2872v1

References in corpus (1)

• Episturmian words: a survey

Cited by in corpus (8)

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• Abelian Combinatorics on Words: a Survey
• Balance properties of Arnoux-Rauzy words
• On the abelian complexity of the Rudin-Shapiro sequence
• Balancedness of Arnoux-Rauzy and Brun words
• Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
• On the abelian complexity of generalized Thue-Morse sequences
• Abelian properties of Parry words