Almost automorphic and bijective factors of substitution shifts

In this article we completely characterise constant length substitution shifts which have a proper almost automorphic factor, or which have a bijective substitution factor such that the factor map is injective on at least one point. Our approach is algebraic: we characterise these dynamical properties in terms of a finite semigroup defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green's <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> -relation and the existence of bijective factors in terms of Green's <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math> -relation. Our results are constructive.