About cube-free morphisms

Abstract

We address the characterization of finite test-sets for cubefreeness of morphisms between free monoids, that is, the finite sets T such that a morphism f is cube-free if and only if f(T) is cube-free. We first prove that such a finite test-set does not exist for morphisms defined on an alphabet containing at least three letters. Then we prove that for binary morphisms, a set T of cube-free words is a test-set if and only if it contains twelve particular factors. Consequently, a morphism f on fa; bg is cube-free if and only if f(aabbababbabbaabaababaabb) is cube-free (length 24 is optimal). Another consequence is an unpublished result of Leconte: A binary morphism is cube-free if and only if the images of all cube-free words of length 7 are cube-free. We also prove that, given an alphabet A containing at least two letters, the monoid of cube-free endomorphisms on A is not finitely generated.