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 : 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. 
James Currie and Narad Rampersad