A string attractor of a string $T[1..|T|]$ is a set of positions $Γ$ of $T$ such that any substring $w$ of $T$ has an occurrence that crosses a position in $Γ$, i.e., there is a position $i$ such that $w = T[i..i+|w|-1]$ and the intersection $[i,i+|w|-1]\cap Γ$ is nonempty. The size of the smallest string attractor of Fibonacci words is known to be $2$. We completely characterize the set of all smallest string attractors of Fibonacci words, and show a recursive formula describing the $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ distinct position pairs that are the smallest string attractors of the $n$th Fibonacci word for $n \geq 7$. Similarly, the size of the smallest string attractor of period-doubling words is known to be $2$. We also completely characterize the set of all smallest string attractors of period-doubling words, and show a formula describing the two distinct position pairs that are the smallest string attractors of the $n$th period-doubling word for $n\geq 2$. Our results show that strings with the same smallest attractor size can have a drastically different number of distinct smallest attractors.

翻译：字符串 $T[1..|T|]$ 的一个字符串吸引子是指 $T$ 的一个位置集合 $Γ$，使得 $T$ 的任意子串 $w$ 都存在一个跨越 $Γ$ 中某个位置的匹配，即存在位置 $i$ 满足 $w = T[i..i+|w|-1]$ 且区间 $[i,i+|w|-1]\cap Γ$ 非空。已知斐波那契词的最小字符串吸引子大小为 $2$。我们完整刻画了斐波那契词的所有最小字符串吸引子集合，并给出了一个递归公式，描述了对于 $n \geq 7$，第 $n$ 个斐波那契词的 $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ 个不同的位置对，这些位置对即为其最小字符串吸引子。类似地，已知倍周期词的最小字符串吸引子大小也为 $2$。我们也完整刻画了倍周期词的所有最小字符串吸引子集合，并给出了一个公式，描述了对于 $n\geq 2$，第 $n$ 个倍周期词的两个不同的位置对，这些位置对即为其最小字符串吸引子。我们的结果表明，具有相同最小吸引子大小的字符串，其不同的最小吸引子数量可能存在巨大差异。