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Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

Published online by Cambridge University Press:  20 September 2016

GREGORY R. MALONEY  and
DAN RUST
GREGORY R. MALONEY
Affiliation:
Newcastle University, Newcastle upon Tyne, NE1 7RU, UK email [email protected]
DAN RUST
Affiliation:
Bielefeld University, Bielefeld 33501, Germany email [email protected]
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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 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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 1086 - 1117
Copyright
© Cambridge University Press, 2016 

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