There are kk-uniform cubefree binary morphisms for all k≥0k≥0

A word is cubefree if it contains no non-empty subword of the form xxxxxx. A morphism h:Σ∗→Σ∗h:Σ∗→Σ∗ is k  -uniform if h(a)h(a) has length k   for all a∈Σa∈Σ. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k≥0k≥0 there exists a k  -uniform cubefree binary morphism. By a result of Leconte, this implies the following stronger result: for all k≥0k≥0 and n≥3n≥3, there exists a k  -uniform nn-power-free binary morphism.