Substitution Tilings Research Articles

Affability of equivalence relations arising from two-dimensional substitution tilings

We will show that an equivalence relation on a Cantor set arising from a two-dimensional substitution tiling by polygons is affable in the sense of Giordano, Putnam and Skau.

When shape matters: deformations of tiling spaces

We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map I from the parameter space of possible shapes of tiles to H of a model tiling space, with values in R. Two tiling spaces that have the same image under I are mutually locally derivable (MLD). When the difference of the images is “asymptotically negligible”, then the tiling dynamics are topologically conjugate, but generally not MLD. For substitution tilings, we give a simple test for a cohomology class to be asymptotically negligible, and show that infinitesimal deformations of shape result in topologically conjugate dynamics only when the change in the image of I is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be topologically weakly mixing. 1991 Mathematics Subject Classification. 37B50, 52C23, 37A20, 37A25, 52C22.

Non-Periodic Rhomb Substitution Tilings that Admit Order n Rotational Symmetry

For each n ∈ ℕ we construct a substitution rule using the set of rhombs with angles (πk)/n. These substitution rules generate a local isomorphism class of tilings closed under rotation of order 2n, and also admit singular tilings fixed under a rotation of order n. The scaling factors for this set of substitution rules includes algebraic numbers of every rank.

Asymptotic arc-components of unimodal inverse limit spaces

We consider the inverse limit space ( I , f ) of a unimodal bonding map f as fixed bonding map. If f has a periodic turning point, then ( I , f ) has a finite non-empty set of asymptotic arc-components. We show how asymptotic arc-components can be determined from the kneading sequence of f. This gives an alternative to the substitution tiling space approach taken by Barge and Diamond [Ergodic Theory Dynamical Systems 21 (2001) 1333].

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.'' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.

The Penrose, Ammann and DA tiling spaces are Cantor set fiber bundles

Anderson and Putnam have recently shown that the space of all tilings of a substitution tiling scheme is a special case of the present author's ‘expanding attractor’. This resurrects an earlier conjecture of the present author in this special case: The tiling space of a substitution tiling space is a fiber bundle over the torus, with Cantor set fiber. If true, this could be important in the classification of these spaces, just as in the one-dimensional case. These spaces were known (in these two disciplines) to be locally the product of a disk and a Cantor set. Farrell and Jones gave an example showing our more inclusive conjecture is false. Here we show that the Penrose, Ammann and ‘stepped plane’ tiling spaces are in fact such bundles. Among dynamicists, the stepped plane tiling space (in special cases) is known as the ‘DA’ attractor of Smale. The formalism of substitution systems provides a Markov partition, and this is certainly part of their intuitive appeal. However, in dynamics such partitions are rare in dimensions greater than 2 and do not occur for the DA attractor. We consider a weaker concept, that omits this requirement. Finally, we construct an example like the Farrell–Jones example, in sufficient detail that it can be visualized as a smooth tiling space, as opposed to the more rigid geometric tiling space.

ET0L SYSTEMS FOR COMPOSITE DODECAGONAL QUASICRYSTAL PATTERNS

Two types of deterministic substitution tilings with 12-fold symmetry and a Pisot number as inflation factor are generated and described in terms of bracketed L-systems. Composition of the inflation rules allows to construct other types of dodecagonal patterns which can be described with the help of ET0L-systems and may be used in order to derive nondeterministic models of quasicrystal structures.

Stochastic L-systems and quasicrystal patterns

Bracketed stochastic Lindenmayer systems (L-systems) are used for the description of certain species of non-deterministic quasicrystal patterns. Several classes of substitution tilings with eight-fold symmetry are introduced. They form the basis for non-deterministic octagonal structures which can be described with stochastic DT0L-systems. Heptagonal structures can also be described with stochastic DT0L-systems, whereas the appropriate scheme for structures with nine-fold symmetry is a stochastic ET0L-system.

The Ordered K-Theory of C/*-Algebras¶Associated with Substitution Tilings

We consider the C *-algebra, A T , constructed from a substitution tiling system which is primitive, aperiodic and satisfies the finite pattern condition. Such a C *-algebra has a unique trace. We show that this trace completely determines the order structure on the group K 0(A T ); a non-zero element in K 0(A T ) is positive if and only if its image under the map induced from the trace is positive.

Regular subdivision in $\mathbf{Z}[\frac{1+\sqrt{5}}{2}

In the ring $\mathbf{Z}[\frac{1+\sqrt{5}}{2}]$, there is a natural subdivision technique analogous to regular subdivision in rational algebraic rings like $\mathbf{Z}[\frac{1}{2}]$. The properties of this subdivision process are developed using the matrix associated to the Fibonacci substitution tiling. These properties are applied to prove some finiteness properties for a discrete group of piecewise-linear homeomorphisms.

Dodecafoam and substitution tilings

We discuss substitution tilings, certain hierarchical, highly structured tilings in the plane or more general geometric spaces. We use “Dodecafoam” as an artful, appealing example.

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

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.

The Local Structure of Tilings and Their Integer Group of Coinvariants

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the $K_0$-group of the groupoid $C^*$-algebra for tilings which reduce to decorations of $\Z^d$. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by $K$-theory for Schr\"odinger operators describing the particle motion in such a tiling.