The branch locus for two dimensional tiling spaces

Abstract

We explore the asymptotic arc components made by the continuous R²-action of translation on two-dimensional nonperiodic substitution tiling spaces. As there is a strong connection between the topology of a tiling space and the tiling dynamics that it supports, the results in this dissertation represent a qualitative approach to the study of tiling dynamics. Our results are the establishment of techniques to isolate and visualize the asymptotic behavior. In a recent paper, Barge, et al, showed the cohomology formed from the asymptotic structure in one-dimensional Pisot substitution tiling spaces is a topological invariant, [BDS]. However, in one dimension there exist only a finite number of asymptotic pairs, whereas there are infinitely many asymptotic leaves in two dimensions. By considering periodic tilings that are asymptotic in more than a half plane we are able to use the stable manifold under inflation and substitution to show there are a finite number of 'directions' of branching. This yields a description of the asymptotic structure in terms of an inverse limit of a branched set in the approximating collared Anderson-Putnam complex. Using rigidity results from [JK], we show the cohomology formed from the asymptotic structure is a topological invariant.