Spectral properties of substitutions on compact alphabets

We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous singular continuous. This allows one construct examples which Koopman operator associated function space has specific spectral components. For abelian bijective substitutions, we a dichotomy result regarding type of diffraction. also first example substitution that countably infinite Lebesgue components Lastly, give non-constant length alphabet gives rise substitutive Delone sets type. extends theory finite with inflation symmetry.