Space tilings and local isomorphism

Radin, Charles; Wolff, Mayhew

doi:10.1007/bf02414073

Cited by 122 publications

(106 citation statements)

References 12 publications

Supporting

Mentioning

104

Contrasting

Unclassified

Order By: Relevance

“…It consists of two steps, namely, establishing a finiteness condition and performing a standard diagonalization procedure; compare [19] and [20].…”

Section: Appendix a Compactness Of Linearly Repetitive Tiling Spacesmentioning

confidence: 99%

“…Hence, one of them, say P, occurs infinitely often. Delete all the tilings from the sequence (T m−1 n ) n∈N which have P n = P. By the Selection Theorem [7], [19], the remaining sequence has a subsequence such that the corresponding setsṖ n k converge with respect to standard Hausdorff metric. Call this sequence (T m n ) n∈N .…”

Section: Appendix a Compactness Of Linearly Repetitive Tiling Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

Damanik¹,

Lenz²

2001

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…It consists of two steps, namely, establishing a finiteness condition and performing a standard diagonalization procedure; compare [19] and [20].…”

Section: Appendix a Compactness Of Linearly Repetitive Tiling Spacesmentioning

confidence: 99%

Section: Appendix a Compactness Of Linearly Repetitive Tiling Spacesmentioning

confidence: 99%

Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

Damanik¹,

Lenz²

2001

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…This means that each individual point x ∈ Λ ∩ B 1/ε (0) may be independently displaced to match Λ ′ ∩ B 1/ε (0), but only within B ε (x). This topology is induced by the Hausdorff metric [18,19]. In general, however, LI classes are not closed.…”

Section: Symmetries Of LI Classes and Hullsmentioning

confidence: 99%

On the Notions of Symmetry and Aperiodicity for Delone Sets

Baake

Grimm

2012

Symmetry

View full text Add to dashboard Cite

Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.

show abstract

“…A key point is that the vague topology on the space X of a uniformly discrete point process is precisely the topology most commonly used in the study of point set dynamical systems [26]. Sometimes this is called the local topology since it implies a notion of closeness that depends on the local configuration of points (as opposed to other topologies that depend only on the long-range average structure of the point set).…”

Section: Point Processesmentioning

confidence: 99%

“…8 has the weighted form: Let g, h ∈ BM c (R d ) and suppose that g * h * μ w 1 is a continuous function on R d . Then for all t ∈ R d , (26) g * h * μ…”

mentioning

confidence: 99%

Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

Deng

Moody

2008

Journal of Geometry and Physics

View full text Add to dashboard Cite

Abstract. The setting is an ergodic dynamical system (X, µ) whose points are themselves uniformly discrete point sets Λ in some space R d and whose group action is that of translation of these point sets by the vectors of R d . Steven Dworkin's argument relates the diffraction of the typical point sets comprising X to the dynamical spectrum of X. In this paper we look more deeply at this relationship, particularly in the context of point processes.We show that there is an R d -equivariant, isometric embedding, depending on the scattering strengths (weights) that are assigned to the points of Λ ∈ X, that takes theWe examine the image of this embedding and give a number of examples that show how it fails to be surjective. We show that full information on the measure µ is available from the weights and set of all the correlations (that is, the 2-point, 3-point, . . . , correlations) of the typical point set Λ ∈ X.We develop a formalism in the setting of random point measures that includes multi-colour point sets, and arbitrary real-valued weightings for the scattering from the different colour types of points, in the context of Palm measures and weighted versions of them. As an application we give a simple proof of a square-mean version of the Bombieri-Taylor conjecture, and from that we obtain an inequality that gives a quantitative relationship between the autocorrelation, the diffraction, and the ǫ-dual characters of typical element of X. The paper ends with a discussion of the Palm measure in the context of defining pattern frequencies.

show abstract

Space tilings and local isomorphism

Abstract: We prove for a large class of tilings that, given a finite tile set, if it is possible to tile Euclidean n-space with isometric copies of this set, then there is a tiling with the 'local isomorphism property'.

Cited by 122 publications

References 12 publications

Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

On the Notions of Symmetry and Aperiodicity for Delone Sets

Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

Contact Info

Product

Resources

About