Ergodic Theory and Dynamical Systems



Tiling spaces are Cantor set fiber bundles


LORENZO SADUN a1 and R. F. WILLIAMS a1
a1 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA (e-mail: sadun@math.utexas.edu)

Abstract

We prove that fairly general spaces of tilings of \mathbb{R}^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in the second author's recent work, and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a \mathbb{Z}^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that (1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), (2) only a finite number of tile types are allowed, and (3) each tile type appears in only a finite number of orientations. The proof is constructive and we illustrate it by constructing a ‘square’ version of the Penrose tiling system.

(Received January 2 2001)
(Revised November 19 2001)