Topology of Tiling Spaces

“…Following Sadun [26,Theorem 1.4], we obtain a tiling from a primitive substitution rule ω on P in the following way. Fix a prototile p ∈ P. Since ω is primitive, there exists n ∈ N sufficiently large such that there exists a translate of p, say t, in the patch ω n (p) whose support is contained in the interior of the support of ω n (p).…”

Section: Nonperiodic Tilings and Their Propertiesmentioning

confidence: 99%

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

Mampusti

Whittaker

2017

Journal of Geometry and Physics

View full text Add to dashboard Cite

Abstract. We introduce a new class of noncommutative spectral triples on Kellendonk's C * -algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using PerronFrobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is θ-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.

show abstract

“…This is an example of a particularly simple hull. In contrast, the hull of the Ammann-Beenker point set, say, is a rather complicated topological space, which locally looks like a product of R 2 with a Cantor set [21,22].…”

Section: Proposition 2 If λ ⊂ R D Is An Flc Set Its Hull X(λ) Is Commentioning

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

Topology of Tiling Spaces

Cited by 110 publications

References 0 publications

Finitely balanced sequences and plasticity of 1-dimensional tilings

Finitely balanced sequences and plasticity of 1-dimensional tilings

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

On the Notions of Symmetry and Aperiodicity for Delone Sets

Contact Info

Product

Resources

About