Topological Bragg Peaks and How They Characterise Point Sets

Kellendonk, Johannes

doi:10.12693/aphyspola.126.497

Cited by 4 publications

(1 citation statement)

References 15 publications

(20 reference statements)

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“…In this case, the corresponding parts of [1,11,42] can be summarised as giving that these three regimes correspond exactly to the situation that Λ is crystallographic, a regular model set, a model set respectively. We refrain from giving precise definitions or proofs but rather refer the reader to [2] for a recent discussion; see [40] as well.…”

Section: Continuous Eigenfunctions and The Maximal Equicontinuous Factormentioning

confidence: 99%

Spectral notions of aperiodic order

Baake¹,

Lenz²

2017

Discrete &Amp; Continuous Dynamical Systems - S

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Section: Continuous Eigenfunctions and The Maximal Equicontinuous Factormentioning

confidence: 99%

Spectral notions of aperiodic order

Baake¹,

Lenz²

2017

Discrete &Amp; Continuous Dynamical Systems - S

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Rokhlin Dimension for Flows

Hirshberg

Szabó

Winter

et al. 2016

Commun. Math. Phys.

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We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C * -algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C * -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C * -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C * -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

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Bragg spectrum, K-theory, and gap labeling of aperiodic solids

Kellendonk

2023

Journal of Mathematical Physics

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The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute the topological Bragg spectrum. We relate the topological Bragg spectrum to topological invariants (Chern numbers) of the solid and to the gap-labeling group, which is the group of possible gap labels for the spectrum of a Schrödinger operator describing the electronic motion in the solid.

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Topological Bragg Peaks and How They Characterise Point Sets

Cited by 4 publications

References 15 publications

Spectral notions of aperiodic order

Spectral notions of aperiodic order

Rokhlin Dimension for Flows

Bragg spectrum, K-theory, and gap labeling of aperiodic solids

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