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Title: Tiling spaces, codimension one attractors and shape
Authors: Clark, Alexander
Hunton, J.
First Published: 11-Oct-2012
Publisher: Electronic Journals Project
Citation: New York Journal Of Mathematics, 2012, 18, pp. 765-796 (32)
Abstract: We establish a close relationship between, on the one hand, expanding, codimension one attractors of diffeomorphisms on closed manifolds (examples of so-called strange attractors), and, on the other, spaces which arise in the study of aperiodic tilings. We show that every such orientable attractor is homeomorphic to a tiling space of either a substitution or a projection tiling, depending on its dimension. We also demonstrate that such an attractor is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the nonorientable case, to a space with a two-to-one covering by such a torus-less-points. This puts considerable constraints on the topology of codimension one attractors, and constraints on which manifolds tiling spaces may be embedded in. In the process we develop a new invariant for aperiodic tilings, which, for 1-dimensional tilings is in many cases finer than the cohomological or K-theoretic invariants studied to date.
ISSN: 1076-9803
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2012, the authors. The file associated with this record is distributed under the Creative Commons “Attribution Non-Commercial No Derivatives” licence, further details of which can be found via the following link:
Appears in Collections:Published Articles, Dept. of Mathematics

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