There are k-uniform cubefree binary morphisms for all k >= 0

TL;DR

This paper proves that for every non-negative integer k, there exists a binary morphism that is both k-uniform and preserves cubefreeness, meaning it maps cubefree words to cubefree words.

Contribution

It establishes the existence of k-uniform cubefree binary morphisms for all k >= 0, filling a gap in combinatorics on words.

Findings

Existence of k-uniform cubefree binary morphisms for all k >= 0

Construction methods for such morphisms

Implications for combinatorics on words and pattern avoidance

Abstract

A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Sigma^* -> Sigma^* is k-uniform if h(a) has length k for all a in Sigma. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k >= 0 there exists a k-uniform cubefree binary morphism.