A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Cech cohomology is related to properties of the fault lines.
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical properties of the space (such as mixing, or the spectrum of the translation operator) are closely related to bulk properties of the individual tilings (such as the diffraction pattern). The topology of the space of tilings, particularly the Cech cohomology, gives information on how the original tiling can be deformed. Tiling spaces can be constructed as inver…
This is a chapter surveying the current state of our understanding of tilings with infinite local complexity. It is intended to appear in the volume {\em Directions in Aperiodic Order}, D. Lenz, J. Kellendonk, and J. Savienen, eds.
We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This description allows to compute its cohomology groups, and prove that it is a model set.
We propose a formalism for tilings with infinite local complexity (ILC), and especially fusion tilings with ILC. We allow an infinite variety of tile types but require that the space of possible tile types be compact. Examples include solenoids, pinwheel tilings, tilings with fault lines, and tilings with infinitely many tile sizes, shapes, or labels. Special attention is given to tilings where the infinite local complexity comes purely from geometry (shears) or comes purely from combinatorics (labels). We examine spectral properties of the invariant measures and define a new notion of complex…
We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent αin terms of the ranks of certain groups which appear in the construction. We give bounds for α. These computations apply to some well known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over \Q. We show that s…
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of $d$-dimensional infinite words: if two $\mathbb{Z}^d$-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent. An analogue theorem is proved for the repetitivity function, which …
This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry.
We define a relative version of tiling cohomology for the purpose of comparing the topology of tiling spaces when one is a factor of the other. We illustrate this with examples, and outline a method for computing the cohomology of tiling spaces of finite type.
Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space of tilings of M, with finite local complexity (relative to some symmetry group Gamma) and closed in the natural topology. Then Omega is the inverse limit of a sequence of compact finite-dimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Gamma. This result extends previous results of Anderson and Putnam, of Ormes, Radin and Sadun, of Bellissard, Benedetti and Gambaudo, and of Gähler. In particular, the construction in this paper is a …
We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion…
We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms, and show that all homeomorphisms are a combination of tiling deformations, translations, and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology, and show that all orbit equivalences are combinations of tiling deformations, transla…
The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the plane with remarkable combinatorial and geometric properties. However, it doesn't have finite local complexity in any usual sense, and therefore we cannot study it with the usual tiling theory. The appeal of the tiling is that all the tiles are conformally regular pentagons. But conformal maps are not allowable under finite local complexity. On the other hand, the tiling can be described completely by its combinatorial data, which rather auto…
A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of Z^2, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.
After investigating by examples the unusual and striking elementary properties of the Penrose tilings and the Arnold cat map, we associate a finite symbolic dynamics with finite grammar rules to each of them. Instead of studying these Markovian systems with the help of set-topology, which would give only pathological results, a noncommutative approximately finite C*-algebra is associated to both systems. By calculating the K-groups of these algebras it is demonstrated that this noncommutative point of view gives a much more appropriate description of the phase space structure of these systems …
A spectral sequence is defined which converges to the Čech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and ePE cohomology groups of the tiling, and the only potentially non-trivial boundary map has a simple combinatorial description in terms of its local patches. Using this spectral sequence, we compute the Čech cohomology of the Euclidean hull of the Penrose tilings.
Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from …
We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obta…
In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in $P$. A coloring $c$ is said to be of low complexity with respect to a rectangle if there exist $m,n\in\mathbb{N}$ and a set $P$ of $n\times m$ rectangular patterns such that $c$ is valid with respect to $P$ and $|P|\leq nm$. Open since it was stated in 1997, Nivat's conjecture states that such a coloring is necessarily periodic. If Nivat's conjecture is true, …
We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space $Ω$ with finite local complexity and study self-maps $F$ that are homotopic to the identity and whose displacements are strongly pattern equivariant (sPE). In place of the familiar rotation number we define a cohomology class $[μ]$. We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poncaré's Theorem: If $[μ]$ is irrational, then $F$ is semi-conjugate to uniform translation on a space $Ω_μ$ of tilings that is homeom…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, …
Let $\Tt$ be an aperiodic and repetitive tiling of $\RM^d$ with finite local complexity. We present a spectral sequence that converges to the $K$-theory of $\Tt$ with $E_2$-page given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of $\Tt$ generalizes the cohomology of the base space of a fibration with local coefficients in the $K$-theory of its fiber. We prove that it is isomorphic to the Čech cohomology of the hull of $\Tt$ (a compactification of the family of its tran…
An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. This construction it rather flexible, so it can be …
We prove that in every compact space of Delone sets in $\mathbb{R}^d$ which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of $\mathbb{R}^d$. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all min…
