Balance and Abelian complexity of the Tribonacci word

Abstract

G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ 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 $\tau$.Comment: 20 pages, 1 figure. This is an extended version of 0904.2872v

Abelian complexity of minimal subshifts

“…Let us also mention that, while the Tribonacci word, the fixed point of the morphism 0 → 01, 0 → 02 and 2 → 0, has as well-behaving subword complexity n → 2n + 1, its Abelian complexity seems to be much more erratic. It is known [29], however, that the range of this complexity is the set {3, 4, 5, 6, 7}.…”

On Collapsing Prefix Normal Words

“…Abelian equivalence has been studied with various generalisations and specifications such as abelian-complexity, k-abelian equivalence, avoidability of (k-)abelian powers and much more (cf. e.g., [5,7,10,8,13,18,20,19] ). The number of occurrences of each letter is captured in the Parikh vector (also known as Parikh image or Parikh mapping) ( [17]): given a lexicographical order on the alphabet, the i th component of this vector is the amount of the i th letter of the alphabet in a given word.…”

On the abelian complexity of the Rudin–Shapiro sequence

“…For example, the abelian complexity functions of some notable sequences, such as the Thue-Morse sequence and all Sturmian sequences, were studied in [11] and [6] respectively. There also many other works devoted to this subject, see [3,9,7,10] and references therein. In the following, we will give the definition of the abelian complexity.…”