arXiv subjects

Lorenzo Sadun

Publications and source records attributed to Lorenzo Sadun.

At least 37 records · Page 2Linked to original sources

arXiv2018

Tiling spaces are Cantor set fiber bundles

We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3) each tile type appears in only a finite number of orientations. The proof is constructive, and we illustrate it by constructing a `square' version of the Penrose tiling system.

math.DS↗

arXiv2018

When size matters: subshifts and their related tiling spaces

We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.

math.DS↗

arXiv2018

Uniqueness and Symmetry in Problems of Optimally Dense Packings

We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples of aperiodic tilings in Euclidean space and a detailed analysis of a new family of examples in the hyperbolic plane. Our goal is to understand qualitative features of such optimum density problems, in particular the appropriate meaning of the uniqueness of solutions, and the role of symmetry in classfying optimally dense packings.

math.MG↗

arXiv2018

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 ${\mathcal I}$ from the parameter space of possible shapes of tiles to $H^1$ of a model tiling space, with values in ${\mathbb R}^d$. Two tiling spaces that have the same image under ${\mathcal I}$ 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 ${\mathcal I}$ is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be topologically weakly mixing.

math.DS↗

arXiv2018

On 2-generator subgroups of SO(3)

We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $π$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $π$ and $π/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.

math.GR↗

arXiv2018

A Simple Geometric Representative for $μ$ of a Point

For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $μ$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p) $ is reducible $\}$, with coefficient -1/4 and appropriate orientation, is our desired geometric representative.

dg-ga↗

arXiv2018

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.

math.DS↗

arXiv2018

Cohomology of Substitution Tiling Spaces

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

math.DS↗

arXiv2018

An Approximation Theorem for Maps Between Tiling Spaces

We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.

math.DS↗

arXiv2018

Homological Pisot Substitutions and Exact Regularity

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.

math.DS↗

arXiv2018

Exact Regularity and the Cohomology of Tiling Spaces

The Exact Regularity Property was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider exact regularity for arbitrary tiling spaces. Let ${T}$ be a $d$ dimensional repetitive tiling, and let $Ω_{T}$ be its hull. If $\check H^d(Ω_{T}, Q) = Q^k$, then there exist $k$ patches whose appearance govern the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling ${T}$ comes from a substitution, then we can quantify that convergence rate. If ${T}$ is also one-dimensional, we put constraints on the measure of any cylinder set in $Ω_{T}$.

math.DS↗

arXiv2018

Quotient Cohomology for Tiling Spaces

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.

math.DS↗

arXiv2018

Fusion: a general framework for hierarchical tilings of $R^d$

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.

math.DS↗

arXiv2018

Fusion tilings with infinite local complexity

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 complexity that applies to ILC tilings.

math.DS↗

arXiv2018

Conjugacies of model sets

Let $M$ be a model set meeting two simple conditions: (1) the internal space $H$ is a product of $R^n$ and a finite group, and (2) the window $W$ is a finite union of disjoint polyhedra. Then any point pattern with finite local complexity (FLC) that is topologically conjugate to $M$ is mutually locally derivable (MLD) to a model set $M'$ that has the same internal group and window as $M$, but has a different projection from $H \times R^d$ to $R^d$. In cohomological terms, this means that the group $H^1_{an}(M,R)$ of asymptotically negligible classes has dimension $n$. We also exhibit a counterexample when the second hypothesis is removed, constructing two topologically conjugate FLC Delone sets, one a model set and the other not even a Meyer set.

math.DS↗

arXiv2018

Tiling deformations, cohomology, and orbit equivalence of tiling spaces

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, translations, and topological conjugacies. These generalize results of Parry and Sullivan to higher dimensions. When the tiling spaces are uniquely ergodic, we show that homeomorphisms (FLC) or orbit equivalences (ILC) are completely parametrized by the appropriate cohomological invariants. We also show that, under suitable cohomological conditions, continuous maps between tiling spaces are homotopic to compositions of tiling deformations and either local derivations (FLC) or factor maps (ILC).

math.DS↗

arXiv2018

Pattern-Equivariant Cohomology with Integer Coefficients

We relate Kellendonk and Putnam's pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with coefficients in any Abelian group. It also provides an easy proof of Kellendonk and Putnam's original theorem relating PE cohomology to the Cech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.

math.DS↗

arXiv2018

Tilings, tiling spaces and topology

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 inverse limits of branched manifolds.

math.DS↗