The paper is concerned with the study of one-dimensional tiling spaces arising from non-primitive rules from the point of view of the topology, dynamics and cohomology. The study is divided into two cases: minimal and non-minimal tiling spaces.
In the minimal case (Sections \(2\) and \(3\)), the authors identify and characterize the so-called tameness property. In particular, as a corollary of this characterization, they deduce that all the aperiodic substitutions are tame. The other main result for the minimal case (Theorem 2.1) allows one to replace a non-primitive substitution with a primitive one if the substitution is minimal. More precisely, if \(\varphi\) is a minimal substitution with non-empty minimal subshift \(X_{\varphi}\), then there exist an alphabet \(\mathcal Z\) and a primitive substitution \(\theta\) on \(\mathcal Z\) such that \(X_{\theta}\) is topologically conjugate to \(X_{\varphi}\). It is worth pointing out that this result is similar to a result from F. Durand [Discrete Math. 179, 89–101 (1998; Zbl 0895.68087)].
The non-minimal case is analyzed in the rest of Sections \(4, 5\) and \(6\). The main result (Theorem 4.5) of Section \(4\) states that if \(\varphi\) is a tame recognizable substitution, then there exist a complex \(\Gamma\) and a map \(f: \Gamma\rightarrow\Gamma\) such that the corresponding tiling space \(\Omega_\varphi\) is homeomorphic to the inverse limit of \(f\). Thus the authors extend to the non-minimal setting the result obtained by J.E. Anderson and I. F. Putnam [Ergodic Theory Dyn. Syst. 18, 509–537 (1998; Zbl 1053.46520)] for primitive substitutions. The authors use the last result to explore, in terms of the cohomology, the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space. The paper finishes with several examples with the aim of illustrating carefully all the machinery developed in the paper.