The Penrose, Ammann and DA tiling spaces are Cantor set fiber bundles

Abstract

Anderson and Putnam have recently shown that the space of all tilings of a substitution tiling scheme is a special case of the present author's ‘expanding attractor’. This resurrects an earlier conjecture of the present author in this special case: The tiling space of a substitution tiling space is a fiber bundle over the torus, with Cantor set fiber. If true, this could be important in the classification of these spaces, just as in the one-dimensional case. These spaces were known (in these two disciplines) to be locally the product of a disk and a Cantor set. Farrell and Jones gave an example showing our more inclusive conjecture is false. Here we show that the Penrose, Ammann and ‘stepped plane’ tiling spaces are in fact such bundles. Among dynamicists, the stepped plane tiling space (in special cases) is known as the ‘DA’ attractor of Smale. The formalism of substitution systems provides a Markov partition, and this is certainly part of their intuitive appeal. However, in dynamics such partitions are rare in dimensions greater than 2 and do not occur for the DA attractor. We consider a weaker concept, that omits this requirement. Finally, we construct an example like the Farrell–Jones example, in sufficient detail that it can be visualized as a smooth tiling space, as opposed to the more rigid geometric tiling space.