Atomic motion and transverse geometry of tiling spaces

Abstract:

The atomic structure of an aperiodic solid, in a d-dimensional physical space, is described by its Hull and its Tiling Space. The Hull can be seen as a foliated space, with leaves given by copies of the physical space. Solids with ``Finite Local Complexity'' (FLC) are the most rigid, have long range order and are described as an inverse limit of CW-complex, called the Anderson-Putnam complex. The Tiling Space can be seen as a transversal to the foliation. For FLC aperiodic solids, the Tiling Space is a Cantor set. The description of the atomic motion will be discussed in terms of the transverse Geometry. In particular the Tiling Space can be treated as the noncommutative analog of a Riemannian manifold and the analog of the Laplace Beltrami operator, called the Pearson Laplacian, can be defined. As a warm up, a simplified model of equilibrium dynamics for ``phason modes'', also called ``flip-flop'', observed in quasicrystals, will be proposed and discussed and compared with the Pearson Laplacian. Some speculative prospect will be proposed in view of using this formalism to represent more complex solids including defects and cracks.