Topological invariants for substitution tilings and their associated $C^\ast$-algebras

Abstract

We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

A spectral sequence for theK-theory of tiling spaces

“…These two results are sufficient to get the K-theory of the hull which, thanks to the Thom-Connes theorem [21], gives also the K-theory of the C * -algebra of the tiling C(Ω) ⋊ R d . However, the construction of the hull through an inverse limit of branched manifolds, initiated by Anderson and Putnam [1] for the case of substitution tilings and generalized in [13] to all repetitive tilings with finite local complexity, suggests a different and more canonical construction. So far however, it is not yet efficient for practical calculations.…”

“…Further examples, and methods of calculation of PV cohomology will be investigated in future research. However, as it turns out (see section 4.1), the PV cohomology is isomorphic to other cohomologies used so far on the hull, such as theČech cohomology [1,63], the group cohomology [27] or the pattern equivariant cohomology [42,43,66]. …”

“…Another important step was performed in 1998 by Anderson and Putnam [1], who proposed to build the tiling space of a substitution tiling through a CW-complex built from the prototiles of a tiling (called prototile space in this paper). The substitution induces a map from this CW-complex into itself and they showed that the inverse limit of such system becomes homeomorphic to the tiling space.…”

“…It started in the 70's with the work of Penrose [57] and Meyer [51] and went on both from an abstract mathematical level and with a view towards applications, in particular to the physics of quasicrystals [45,46]. It contains an important subclass of the class of substitution tilings that was reinvestigated in the 90's by Anderson and Putnam in [1], and also the class of tilings obtained by the cut-and-projection method, for which a comprehensive study by Hunton, Kellendonk and Forrest can be found in [28], and more generaly it contains the whole class of quasiperiodic tilings, which are models for quasicrystals. The Ammann aperiodic tilings are examples of substitution tilings (see [33] chapter 10, Ammann's original work does not appear in the literature).…”

Canonical symbolic dynamics for one-dimensional generalized solenoids

“…For case (2), let k be the smallest positive integer such that [f kl (a), f kl (b)] contains a vertex. It follows from the Nonfolding Axiom that …”