Tribonacci词的平衡性和阿贝尔复杂性
Balance and Abelian complexity of the Tribonacci word
摘要：
G. Rauzy证明了态射$τ: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$所产生的Tribonacci最小子移位在测量理论上共轭于$R^2$中紧集上的三个分形域的交换，每个域都由相同的向量模格平移。本文研究了Tribonacci字$t$的阿贝尔复杂度AC（n），它是$τ$的唯一不动点.我们表明，$AC(n)\in {3,4,5,6,7}$为每个$n\geq 1$，这五个值的每一个假设。我们的证明依赖于这样一个事实，即Tribonacci词是2平衡的，即，对于长度相等的$t$的所有因子$U$和$V$，以及对于每个字母$a \in {0,1,2}$，$a$在$U$中出现的次数和$a$在$V$中出现的次数相差至多2。虽然这一结果在几篇论文中被宣布，但据我们所知，还没有发表过这一事实的证据。我们给出了$t$的2-平衡性质的两个非常不同的证明。第一个使用的词组合属性的生成态射，而第二个利用$τ$的关联矩阵的频谱特性。
英文摘要：
G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $τ: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ 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 $τ$. 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 $τ$.
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