  We develop a systematic approach to continuous substitutions on compact
Hausdorff alphabets. Focussing on implications of irreducibility and
primitivity, we highlight important features of the topological dynamics of
their subshifts. We then reframe questions from ergodic theory in terms of
spectral properties of a corresponding substitution operator. This requires an
extension of standard Perron--Frobenius theory to the setting of Banach
lattices. As an application, we identify computable criteria that guarantee
quasi-compactness of the substitution operator and hence unique ergodicity of
the associated subshift.
