A \emph{morphism} is a mapping that transforms words through letter-wise substitution, where each symbol is consistently replaced by a fixed word. In the field of combinatorics on words, one topic that has attracted considerable attention is the characterization of morphisms that preserve specific properties, such as overlap-freeness, square-freeness, lexicographic order, and primitivity. Continuing this direction, we initiate the study on \emph{occurrence-preserving morphisms}, which address the following fundamental question: given a morphism $φ$, two words $u$ and $v$, and $k \geq 1$, under what conditions does the number of occurrences of $u$ in $v$ equal the number of occurrences of $φ^k(u)$ in $φ^k(v)$? To answer this question, we introduce the notion of \emph{interference-free morphisms}, examine their properties, develop an efficient algorithm for deciding interference-freeness, and uncover a connection to \emph{recognizable morphisms}. We then present a precise characterization of occurrence-preserving morphisms in terms of interference-freeness. As applications of our characterization, we first show that there exists a bijection between the starting positions of the occurrences of $u$ in $v$ and those of $φ^k(u)$ in $φ^k(v)$. We then apply the characterization to the Fibonacci and Thue-Morse words to identify their \emph{minimal unique substrings~(MUSs)}. Finally, we exploit the connection between MUSs and \emph{net occurrences} to simplify existing proofs on net occurrences in these words.

翻译：一个\emph{态射}是一种通过字母替换变换单词的映射，其中每个符号被一致地替换为一个固定单词。在词组合学领域中，一个备受关注的主题是刻画保持特定性质（如无重叠性、无平方性、字典序和本原性）的态射。延续这一方向，我们首次研究了\emph{保持出现次数的态射}，其针对以下基本问题：给定一个态射$φ$、两个单词$u$和$v$以及$k \geq 1$，在什么条件下$v$中$u$的出现次数等于$φ^k(v)$中$φ^k(u)$的出现次数？为回答此问题，我们引入了\emph{无干扰态射}的概念，考察了其性质，开发了一种高效判定无干扰性的算法，并揭示了其与\emph{可识别态射}的联系。随后，我们给出了保持出现次数的态射在无干扰性意义下的精确刻画。作为该刻画的应用，我们首先证明了$v$中$u$的出现起始位置与$φ^k(v)$中$φ^k(u)$的出现起始位置之间存在双射。然后，我们将该刻画应用于斐波那契词和Thue-Morse词，以识别它们的\emph{最小唯一子串（MUSs）}。最后，我们利用MUSs与\emph{净出现次数}之间的联系，简化了这些词中净出现次数的现有证明。