On Occurrence-Preserving Morphisms

TL;DR

This paper introduces and characterizes occurrence-preserving morphisms in combinatorics on words, providing algorithms and applications to well-known sequences like Fibonacci and Thue-Morse words.

Contribution

It defines interference-free morphisms, develops an algorithm for their detection, and characterizes occurrence-preserving morphisms, with applications to minimal unique substrings and net occurrences.

Findings

Established a connection between occurrence-preserving and recognizable morphisms.

Provided an efficient algorithm for deciding interference-freeness.

Applied the characterization to Fibonacci and Thue-Morse words to identify minimal unique substrings.

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 $\phi$, 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 $\phi^k(u)$ in $\phi^k(v)$? 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 $\phi^k(u)$ in $\phi^k(v)$. 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.