发表机构
University of Catania; Univ Rouen Normandie, INSA Rouen Normandie, Université Le Havre Normandie, Normandie Univ; University of Palermo（卡塔尼亚大学; 鲁昂诺曼底大学，鲁昂国立应用科学学院，勒阿弗尔诺曼底大学，诺曼底大学; 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
本文刻画有限 Thue–Morse 字的最小字符串吸引子，明确 $n\geq6$ 时最小吸引子的数量与结构，还给出因子覆盖的规范约束，证明四位置吸引子的测试条件及连续代的必要数量。
AI 中文摘要
字符串吸引子为表示一个字的完整因子结构提供了紧凑方式：若每个不同因子都至少有一个出现位置穿过所选位置集中的一个位置，则该位置集即为吸引子。尽管若干经典字族的最小吸引子规模已明确，但描述所有最优吸引子通常困难得多，因为这需要理解所有因子出现的几何结构，而非仅构造单个最优解。本文针对有限 Thue–Morse 字给出了完整描述。早期工作证明，对于所有阶数 $n\geq4$，4 个位置是必要且充分的，但所有最小吸引子的集合仍未知。对于每个 $n\geq6$，令 $h=2^{n-3}$，本文证明最小吸引子恰好是两个反射族，其中四个偏移量独立选自 $\{0,1\}$，因此对于每个 $n\geq6$，恰好有 32 个最小吸引子。初始情况特殊：$t_5$ 有 40 个最小吸引子，$t_4$ 有 87 个。本文还独立于最优性研究吸引子条件，对于每个 $n\geq5$，刻画了 $t_n$ 的包含极小因子覆盖的完全反链，其恰好由 $aa$、$bb$ 的覆盖以及每一代 $t_m$（$4\leq m\leq n$）的 8 个最小唯一子串组成，因此恰好有 $8n-22$ 个规范约束，形成任意基数吸引子的无冗余精确证书。当限制为四位置集时，该线性规模系统退化为常数规模：只需测试来自三个连续代的 24 个最小唯一子串，其中 16 个已能强制得到两个最优族。本文还证明，在该自然连续代层次中，三个代是必要的。
英文摘要
String attractors provide a compact way of representing the complete factor structure of a word: a set of positions is an attractor if every distinct factor has at least one occurrence crossing one of the selected positions. Although the minimum attractor size is known for several classical families of words, describing \emph{all} optimal attractors is typically much more difficult, since it requires understanding the geometry of all factor occurrences rather than constructing a single optimal solution. We give a complete description for the finite Thue--Morse words. Earlier work proved that four positions are necessary and sufficient for every order $n\geq 4$, but the collection of all smallest attractors remained unknown. For every $n\geq 6$, writing $h=2^{n-3}$, we prove that the smallest attractors are exactly the two reflected families where the four offsets are chosen independently from $\{0,1\}$. Hence there are exactly $32$ smallest attractors for every $n\geq6$. The initial cases are genuinely exceptional: $t_5$ has $40$ smallest attractors and $t_4$ has $87$. We also study the attractor condition independently of optimality. For every $n\geq5$, we characterize the complete antichain of inclusion-minimal factor coverages of $t_n$. It consists precisely of the coverages of $aa$, $bb$, and the eight minimal unique substrings of every generation $t_m$, $4\leq m\leq n$. Thus there are exactly $8n-22$ canonical constraints, forming an irredundant exact certificate for attractors of arbitrary cardinality. When attention is restricted to four-position sets, this linear-size system collapses to a constant one: it is enough to test the $24$ minimal unique substrings coming from three consecutive generations, and sixteen of these already force the two optimal families. We also show that three generations are necessary within this natural consecutive-generation hierarchy.
