We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a universal version of the Anderson-Putnam complex. We show that, under a certain condition on the substitutions, the projective limit of this universal Anderson-Putnam complex is isomorphic to the tiling space, and we introduce a simplified universal Anderson-Putnam complex that can be used to compute Cech cohomology. We then use this simplified complex to place bounds on the rank of the first cohomology group of a one-dimensional substitution tiling space in terms of the number of tiles.

ABSTRACT. We study the topology and dynamics of subshifts and tiling spaces associated to nonprimitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterisation of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution.We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain CW complex under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger that theČech cohomology of the tiling space alone.1. PRELIMINARIES 1.1. Outline. The goal of this work is to study one-dimensional tiling spaces arising from nonprimitive substitution rules, in terms of the topology, dynamics, and cohomology. This study naturally divides into two cases: the case where the tiling space is minimal, and the case where it is non-minimal. The minimal case is treated in Section 2, where we identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. In particular, all aperiodic substitutions will be seen to be tame. By aperiodic, we mean that the subshift of the substitution has no periodic orbits. The first main result is Theorem 2.9, which gives a characterisation of tameness. This theorem is used to prove the following result.Theorem 2.1. Let ϕ be a minimal substitution with non-empty minimal subshift X ϕ . There exists an alphabet Z and a primitive substitution θ on Z such that X θ is topologically conjugate to X ϕ . This is similar to, but slightly stronger than, a result from the section on Open problems and perspectives (Section 6.2) of [10].Examples and applications based on this section are then given in Section 3.

We describe a computer algorithm that searches for substitution rules on a set of triangles, the angles of which are all integer multiples of π/n. We find new substitution rules admitting 7-fold rotational symmetry at many different inflation factors.

Discrete Algorithms International audience A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with elevenfold symmetry appearing in the literature.