Another Solution to the Thue Problem of Non-Repeating Words

TL;DR

This paper presents a new morphism that preserves four non-repeating word properties, solving an open problem and expanding understanding of morphisms in combinatorics on words.

Contribution

It provides the first known morphism that preserves all four key non-repeating properties and proves the non-existence of lower-rank morphisms with these properties.

Findings

Existence of a morphism preserving all four properties

No morphisms with these properties exist at lower rank

Positive solution to the open problem of weakly squarefree morphisms

Abstract

In this work we consider morphisms that preserve well-known non-repeating properties: squarefreeness, cubefreeness, overlap-freeness and weak squarefreeness. Up to the present moment only the morphisms preserving three out of four non-repeating properties have been known. The problem of the existence of weakly squarefree morphisms was open. The essential result of this work is the positive solution to this problem. An example of the morphism preserving all four properties is provided. Also, it is proved that there are no morphisms with the same properties and a lower rank.