AI 中文总结
本文通过显式形态$\sigma_a$揭示了双周期斐波那契词族中替换频率与组合不变量的对偶性关系。
AI 中文摘要
本文研究了由指令序列表(a,b,a,b,…)生成的双周期斐波那契词族$\mathfrak{F}^{(a,b)}$的自然对偶性。$\mathfrak{F}^{(a,b)}$和$\mathfrak{F}^{(b,a)}$通过显式形态$\sigma_a:0\mapsto 0^a1,1\mapsto 0$相互关联，建立了它们之间的精确替换对应关系。我们计算了精确的字母频率，为每个字母给出了完整的返回词描述，证明了存在任意长的回文前缀，并确定了斜率$\theta^{(a,b)}$的连分数展开。这些发现揭示了几个不变量中看似不对称性实际上是由形态$\sigma_a$诱导的长度再分布机制统一引起的。
英文摘要
In this paper, a natural duality on the family of biperiodic Fibonacci words $\mathfrak{F}^{(a,b)}$ generated by the directive sequence $(a,b,a,b,\dots)$. Both of $\mathfrak{F}^{(a,b)}$ and $\mathfrak{F}^{(b,a)}$ are related by the explicit morphism $σ_a:0\mapsto 0^a1$, $1\mapsto 0$, establishing a precise substitutional correspondence between them. We compute the exact letter frequencies, give a complete description of the return words for each letter, prove the existence of arbitrarily long palindromic prefixes, and determine the continued fraction expansion of the slope $θ^{(a,b)}$. These findings reveal that the apparent asymmetry in several invariants arises uniformly from the length-redistribution mechanism induced by the morphism $σ_a$.
Comments16 pages
