An uncountable set of tiling spaces with distinct cohomology

We generalise the notion of a Barge–Diamond complex, in the one-dimensional case, to any mixed system of tiling substitutions. This gives a way of describing the associated tiling space as an inverse limit of Barge–Diamond complexes. We give an effective method for calculating the Čech cohomology of the tiling space via an exact sequence relating the associated sequence of substitution matrices and certain subcomplexes appearing in the approximants. As an application, we show that there exists a system of three substitutions on two letters which exhibit an uncountable collection of minimal tiling spaces with distinct isomorphism classes of Čech cohomology.