Tiling spaces are Cantor set fiber bundles. (English) Zbl 1038.37014

Sadun, Lorenzo; Williams, R. F.

Ergodic Theory Dyn. Syst. 23, No. 1, 307-316 (2003).

The authors consider tiling systems \(P\) of \(\mathbb{R}^d\) for which (1) the tiles are polyhedra which meet face to face, (2) only a finite number of tile types are used, (3) the tiling space is a closed, nonempty, translation-invariant subset of the space of all tilings that can be formed by these tiles.
The main result is that such a tiling space is a fiber bundle over the torus with totally disconnected fiber. This is a piece of the larger issue on the global structure of an expanding attractor. The work of J. E. Anderson and I. F. Putnam [Ergodic Theory Dyn. Syst. 18, 509–537 (1998; Zbl 1053.46520)] on substitution tiling systems to attractors, and the work of R. F. Williams [Publ. Math., Inst. Hautes Étud. Sci. 43, 169–203 (1973; Zbl 0279.58013)] analyze the structure of attractors for discrete dynamical systems. The paper gives the global structure of the tiling systems described above. Most of the work is in showing that these tiling systems are homeomorphic to rational tiling systems, and the proof is illustrated using the Penrose tiling.
It is also shown that a tiling system as described above is homeomorphic to the \(d\)-fold suspension of a \(\mathbb{Z}^d\) subshift. The proof mainly involves showing that the tiling system is topologically conjugate to a square-type tiling system. The details of the proof are again illustrated on the Penrose tiling.

Reviewer: Aimee Johnson (Swarthmore)

MSC:

37B50	Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B10	Symbolic dynamics
37C70	Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

tiling systems; fiber bundle; expanding attractor; Penrose tiling

Citations:

Zbl 0279.58013; Zbl 1053.46520