A uniform cube-free morphism is k-power-free for all integers k ≥ 4

L.A.M.F.A., UMR 7352 du CNRS, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France

^* Corresponding author: wlazinski@u-picardie.fr

Accepted: 4 December 2017

Abstract

In the study of k-power-free morphisms, the case of 3-free-morphisms, i.e., cube-free morphisms, often differs from other k-power-free morphisms. Indeed, cube-freeness is less restrictive than square-freeness. And a cube provides less equations to solve than any integer k ≥ 4. Anyway, the fact that the image of a word by a morphism contains a cube implies relations that, under some assumptions, allow us to establish our main result: a cube-free uniform morphism is a k-power-free morphism for all integers k ≥ 4.