New York Journal of Mathematics
Volume 18 (2012) 765-796


Alex Clark and John Hunton

Tiling spaces, codimension one attractors and shape

view    print

Published: October 11, 2012
Keywords: Aperiodic tilings, tiling spaces, expanding attractors, shape theory.
Subject: Primary: 37D45; secondary: 37B50, 37E30, 52C22, 55P55

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.


The University of Leicester funded study leave for both authors during the course of this research.

Author information

Department of Mathematics, University of Leicester, University Road, Leicester, UK