Tiling Spaces, Codimension One Attractors and Shape

Abstract

We show that any codimension one hyperbolic attractor of a diffeomorphism of a (d+1)-dimensional closed manifold is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the non-orientable case, to a space with a 2 to 1 covering by such a torus-less-points. Furthermore, we show that each orientable attractor is homeomorphic to a tiling space associated to an aperiodic tiling of Rd, but that the converse is generally not true. This work allows the definition of a new invariant for aperiodic tilings, in many cases finer than the cohomological or K-theoretic invariants studied to date.

References in corpus (2)

• Tiling Spaces are Inverse Limits
• Torsion in Tiling Homology and Cohomology

Cited by in corpus (7)

• Beyond primitivity for one-dimensional substitution subshifts and tiling spaces
• An uncountable set of tiling spaces with distinct cohomology
• Computations for symbolic substitutions
• Perspectives on Kuperberg flows
• Grout: A 1-Dimensional Substitution Tiling Space Program
• Aperiodicity, rotational tiling spaces and topological space groups
• The homology core and invariant measures