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

A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Σ * → Σ * is k-uniform if h(a) has length k for all a ∈ Σ. 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.