Automaticity of uniformly recurrent substitutive sequences

We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of an underlying purely substitutive sequence. This gives an answer to a recent question of Allouche, Dekking and Queffélec in the uniformly recurrent case. We also show that the same criterion characterises automaticity of minimal substitutive systems.