Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals

Repetitive Delone Sets and Quasicrystals

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is called repetitive if there is a function M(T) such that every ball of radius M(T)+T contains a copy of each kind of patch of radius T that occurs in the set. This is equivalent to the minimality of an associated topological dynamical system with R^n-action. There is a lower bound for M(T) in terms of N(T), namely N(T) = O(M(T)^n), but no general upper bound. T…

Discontinuities of the integrated density of states for random operators on Delone sets

Despite all the analogies with "usual random" models, tight binding operators for quasicrystals exhibit a feature which clearly distinguishes them from the former: the integrated density of states may be discontinuous. This phenomenon is identified as a local effect, due to occurrence of eigenfunctions with bounded support.

高维傅里叶准晶体从李-杨品种

Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

本文提出了一种在任意维度中构造具有单位质量的傅里叶准晶体的方法，利用复代数簇推广了李-扬多项式定义的超曲面，并证明其具有有限交集的性质。

Linear zero mode spectra for quasicrystals

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of linear zero mode spectra, for an aperiodic bar-joint framework $G$, that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the limit spectrum ${\bf L}_{lim}(G, a)$ is defined in this way, as a subset of the reciprocal space for a reference basis $a$ of the ambient space. A smaller spectrum, the slippage spectrum ${\bf L…

Zero measure Cantor spectra for continuum one-dimensional quasicrystals

We study Schrödinger operators on $\R$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated b…

Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.