Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

• Published in: Ergodic theory and dynamical systems Vol. 38; no. 3; pp. 1086 - 1117
• Main Authors: MALONEY, GREGORY R., RUST, DAN
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.05.2018
• Subjects: Invariants; Original Article; Quotients; Subspaces; Substitutes; Tiling
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2016.58

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive 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 characterization 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 finite graph 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 than the Čech cohomology of the tiling space alone.