Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Maloney, Gregory R. and Rust, Dan (2018). Beyond primitivity for one-dimensional substitution subshifts and tiling spaces. Ergodic Theory and Dynamical Systems, 38(3) pp. 1086–1117.

DOI: https://doi.org/10.1017/etds.2016.58

Abstract

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 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 the Čech cohomology of the tiling space alone.

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