Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

TL;DR

This paper explores the topology and dynamics of non-primitive one-dimensional substitution subshifts and tiling spaces, introducing the concept of tameness to prevent pathological behaviors and extending key results to non-minimal cases.

Contribution

It introduces the property of tameness for non-primitive substitutions, characterizes it, and extends known results about minimal and primitive substitutions to the non-minimal setting.

Findings

Most pathological behaviors are prevented under tameness.

Subshifts of minimal substitutions are conjugate to primitive ones.

Tiling spaces are homeomorphic to inverse limits of finite graphs.

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 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 graphx 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 \v{C}ech cohomology of the tiling space alone.