On Occurrence-Preserving Morphisms

2026-05-18

AI Summary

This study investigates the conditions under which the frequency of a given substring remains invariant under iterative application of a morphism—that is, when two words continue to exhibit equal occurrences of a specific factor after morphic iteration. We introduce and rigorously characterize occurrence-preserving morphisms for the first time, establishing their equivalence with interference-free morphisms and uncovering a bijective structure governing the positions of substring occurrences. Leveraging tools from combinatorics on words and formal language theory, we devise an efficient algorithm to decide this property. Applying our framework to the Fibonacci and Thue–Morse words, we successfully identify their minimal unique substrings (MUSs) and provide a streamlined proof of classical results concerning net occurrences.

Abstract

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.

Problem

• occurrence-preserving morphisms
• combinatorics on words
• morphisms
• word occurrences
• interference-free morphisms

Innovation

• occurrence-preserving morphisms
• interference-free morphisms
• recognizable morphisms
• minimal unique substrings
• combinatorics on words