Generalized $\beta$-expansions, substitution tilings, and local finiteness

Frank, Natalie Priebe; Robinson, E. Arthur

doi:10.1090/s0002-9947-07-04527-8

Cited by 37 publications

(45 citation statements)

References 7 publications

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“…The b3 substitution has been thoroughly studied from the geometrical point of view, see eg. [7]. One can show that under the b3 substitution, the heights are multiplied by 3, along with a sign change, giving:…”

Section: B Generalization To Other Aperiodic Chainsmentioning

confidence: 99%

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

et al. 2017

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We present exact solutions for some eigenstates of hopping models on one and two dimensional quasiperiodic tilings and show that they are "critical" states, by explicitly computing their multifractal spectra. These eigenstates are shown to be generically present in 1D quasiperiodic chains, of which the Fibonacci chain is a special case. We then describe properties of the ground states for a class of tight-binding Hamiltonians on the 2D Penrose and Ammann-Beenker tilings. Exact and numerical solutions are seen to be in good agreement.In quasicrystals, the arrangement of atoms is nonperiodic yet long-range ordered. This type of quasiperiodic order is best illustrated by considering tilings. These are structures obtained by packing a small number of basic elements, or tiles, according to certain specified rules. Quasiperiodic tilings can exhibit non-crystallographic symmetries: in this article we will focus on the AmmannBeenker tiling ( fig. (1)) that has a eight-fold orientational symmetry, and on the celebrated Penrose tiling which has a five-fold symmetry.Single-electron properties of 1D quasicrystals are rather well understood: many spectral properties of quasiperiodic chains are known, and the eigenstates are also well characterized. In contrast, even the simplest models of 2 and 3 dimensional quasicrystals have resisted theoretical investigations. Consider for example the Ammann-Beenker tiling ( fig. (1)), and a tightbinding model for non-interacting electrons hopping between nearest neighbor vertices on this tiling:Although this Hamiltonian is simple, no solutions were known for any of its eigenstates, apart from trivial confined eigenstates at the middle of the spectrum [17]. Taking their cue from periodic crystals, where Bloch states are given by ψ k (r) = u k (r)e ik.r with u a periodic function, Kalugin and Katz [15] proposed that the ground state of the model (1) be given by a product of two factors, namelyIn this expression, κ is a real constant (note that in [15] the notation λ = exp(2κ) is used). The pre-exponential factor C(i) of the ansatz (2) is a quasiperiodic function, a site-dependent amplitude which depends only on the arrangement of the atoms around the site i. In other words, C(i) C(j) if the arrangement of the atoms around site i matches the arrangement of the atoms around site j out to a large distance. The non-local height field h(i) in the exponential depends on the geometrical properties of the tiling. We will refer to eigenstates of this form henceforth as Sutherland-Kalugin-Katz -or SKK eigenstates.In this article, we consider generalizations of the results of Kalugin and Katz to other tight-binding models. We consider first the relatively simple case of models on 1D quasiperiodic chains, and show that they admit eigenstates of a form similar to that given in Eq. (2). We next consider the 2D case, for the Ammann-Beenker and Penrose tilings. and we show that the ground states continue to have the SKK structure even when the Hamiltonian in Eq.(1) is generalized to include onsite potenti...

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Section: B Generalization To Other Aperiodic Chainsmentioning

confidence: 99%

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

et al. 2017

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“…We now construct the dD period doubling structures by straightforward extension of the one-dimensional substitution (inflation, morphism) rule (5) to d dimensions (cf. Frank & Robinson, 2008):…”

Section: Substitutionmentioning

confidence: 99%

Multidimensional period doubling structures

2016

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“…Substitution tilings (with the assumptions on Q stated above) with FLC have the rigidity property, but rigidity does not imply FLC. The following substitution tiling by Frank and Robinson[13] demonstrates this. (See also[1, Ex.…”

mentioning

confidence: 93%

“…For dynamical properties of the latter, we refer the reader to [37,11]. First examples of ILC substitution tilings were constructed by Danzer [9] and Kenyon [18], and a large class of such tilings was studied by Frank and Robinson [13]. More recently, investigation of ILC tilings appeared in the work of Frank and Sadun [14,16], see also [12], in a general framework of "fusion tilings".…”

Section: Introductionmentioning

confidence: 99%

“…The former property is satisfied, in particular, for the pinwheel tiling. It is an open question whether for tilings with fault lines, like in [13], the corresponding point sets are almost linearly repetitive. We should note that the results of Frank and Sadun [16] on unique ergodicity are more general and cover substitution systems of both kinds; however, the framework that we develop is better suited for our purposes in the sequel.…”

Section: Introductionmentioning

confidence: 99%

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On substitution tilings and Delone sets without finite local complexity

Lee¹,

Solomyak²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences.In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

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Generalized $\beta$-expansions, substitution tilings, and local finiteness

Cited by 37 publications

References 7 publications

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

Multidimensional period doubling structures

On substitution tilings and Delone sets without finite local complexity

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