发表机构
University of Catania; University of Palermo（卡塔尼亚大学; 巴勒莫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明纯态射正则外标准词的有限逼近词中净出现次数呈尖锐二分类：当末指数为1时恰为三次，否则为两次，将斐波那契现象推广至更广框架。
AI 中文摘要
最近已证明有限斐波那契词恰好包含三次净出现——即重复因子的一次出现，其单字母左、右扩展是唯一的。这个出乎意料的小常数引出一个自然问题：它是斐波那契递推所特有的，还是更广泛的态射现象的一部分？我们研究了字母表$\{0,1,\ldots,d-1\}$上由正整数$e_0,\ldots,e_{d-1}$决定的一族词。对每个字母$a$，设$L_a$为固定$a$并将每个其他字母$b$映射为$ab$的态射；我们考虑由$\mu=L_0^{e_0}\cdots L_{d-1}^{e_{d-1}}$生成的有限逼近词$S_m=\mu^m(0)$。这些词是此处考虑的纯态射正则外标准族的周期对齐有限逼近词。我们证明了一个尖锐的二分类：对每个$m\ge2$，当$e_{d-1}=1$时$S_m$恰好有三次净出现，当$e_{d-1}\ge2$时恰好有两次；初始逼近词也被完全分类。证明结合了回文前缀、返回词因子分解和重叠净出现覆盖。作为推论，当所有指数都等于1——即标准$d$-波那契情形——每个非初始周期对齐有限逼近词恰好有三次净出现，将斐波那契现象置于更广泛的外标准框架中。
英文摘要
Finite Fibonacci words have recently been shown to contain exactly three net occurrences---occurrences of repeated factors whose one-letter left and right extensions are unique. This unexpectedly small constant raises a natural question: is it peculiar to the Fibonacci recurrence, or part of a broader morphic phenomenon? We study a family over the alphabet $\{0,1,\ldots,d-1\}$ determined by positive integers $e_0,\ldots,e_{d-1}$. For each letter $a$, let $L_a$ be the morphism that fixes $a$ and maps every other letter $b$ to $ab$; we consider the finite approximants $S_m=μ^m(0)$ generated by $μ=L_0^{e_0}\cdots L_{d-1}^{e_{d-1}}$. These words are the period-aligned finite approximants of the purely morphic regular epistandard family considered here. We prove a sharp dichotomy: for every $m\ge2$, $S_m$ has exactly three net occurrences when $e_{d-1}=1$, and exactly two when $e_{d-1}\ge2$; the initial approximant is also completely classified. The proof combines palindromic prefixes, return-word factorizations, and overlapping net-occurrence covers. As a consequence, when all exponents are equal to one---the standard $d$-bonacci case---every noninitial period-aligned finite approximant has exactly three net occurrences, placing the Fibonacci phenomenon in a wider epistandard framework.
