Daniel Lenz

Daniel Lenz
Friedrich Schiller University Jena | FSU · Department of Mathematics

Dr

About

176
Publications
9,501
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3,399
Citations
Additional affiliations
October 2008 - present
Friedrich-Schiller-Universität Jena
August 2007 - April 2008
Rice University
Position
  • Visiting associate professor
January 2003 - present
California Institute of Technology

Publications

Publications (176)
Article
We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple...
Chapter
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima’s ergodic theorem for the harmonic functions in the domain of the \( L^{p} \) generator. Secondly we prove analogues of Yau’s and Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain \(...
Preprint
We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr and Bochner type almost periodicity, and similar conditions, and...
Preprint
Full-text available
We consider continuous $\mathrm{SL}(2,\mathbb{R})$ valued cocycles over general dynamical systems and discuss a variety of uniformity notions. In particular, we provide a description of uniform one-parameter families of continuous $\mathrm{SL}(2,\mathbb{R})$ cocycles as $G_\delta$-sets. These results are then applied to Schr\"odinger operators with...
Article
We study mean equicontinuous actions of locally compact σ-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and provide a characterization of mean equicontinuity of an action via properties of its product. This characterizati...
Article
We show that a generic quasi-periodic Schrödinger operator in L2(ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely...
Chapter
In this section we prove an analogue to Cheeger’s famous theorem on Riemannian manifolds. This result relates an isoperimetric constant, called the Cheeger constant, to the bottom of the spectrum.
Chapter
In Section 4.4 we study the bottom of the essential spectrum of 𝐿. The essential spectrum is the complement in the spectrum of the isolated eigenvalues of finite multiplicity.
Chapter
In this chapter we discuss key concepts in the spectral geometry of infinite graphs. We first introduce in Section 1.1 the setting and the main objects of study found throughout the remainder of the book.
Chapter
Any semigroup coming from an operator associated to a Dirichlet form is positivity preserving. In this section, we will show that if the operator comes from a Dirichlet form which is associated to a connected graph, then the semigroup is positivity improving.
Chapter
In this chapter we study a phenomenon called stochastic completeness. At a basic level, this phenomenon concerns the heat equation on ℓ∞(X) and the preservation of heat for a graph b over (𝑋, 𝑚).We will give a variety of perspectives on this property.
Chapter
The topic presented in this chapter is recurrence. This concept can be studied via probability, potential theory and operator theory and has interpretations in each context.
Chapter
In this chapter we investigate what it means for a graph to have relatively few edges. This leads to the notions of weakly sparse, approximately sparse and sparse graphs, as well as graphs which satisfy a strong isoperimetric inequality.
Chapter
In this chapter we look at consequences of lower bounds on the measure m for a graph (𝑏, 𝑐) over a discrete measure space (𝑋, 𝑚). We formulate the lower bound assumptions in two ways.
Chapter
In this chapter we introduce the notion of an intrinsic metric. Section 11.1 is devoted to definitions and motivations. An important class of examples are so-called path metrics, which we discuss in Section 11.2. In this section we prove a Hopf–Rinow theorem, which characterizes metric completeness.
Chapter
The concept of a graph is one of the most fundamental mathematical concepts ever conceived. Graphs inherently appear in many branches of mathematics and natural sciences.
Chapter
In this chapter we discuss a class of graphs whose geometry has a weak spherical symmetry.We first introduce the notion of spherical symmetry that we wish to study and give several examples.
Chapter
The uniqueness of self-adjoint and Markov extensions of a symmetric operator on a Hilbert space is a classical topic in operator theory. In this chapter we consider these problems from the viewpoint of solutions to equations involving the Laplacian and the viewpoint of the domains of both the operators and the forms.
Chapter
In this chapter we extend the theory of the key concepts introduced in the previous chapter. In particular, we collect various tools that are needed at later parts of the book and provide further conceptual insights.
Chapter
In this chapter we present a volume growth criterion for stochastic completeness. More specifically, we show that the measure of finite balls defined with respect to an intrinsic metric must growsuperexponentially in order for a graph to be stochastically incomplete.
Chapter
The key tool for all of these results are variants of the Caccioppoli inequality which are established in Section 12.1. Roughly speaking, such inequalities allow us to estimate the energy of u times a cutoff function by u times the energy of the cutoff function.
Preprint
Full-text available
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^...
Preprint
Full-text available
We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear op...
Article
We compute the deficiency spaces of operators of the form $H_A{\hat {\otimes }} I + I{\hat {\otimes }} H_B$ , for symmetric $H_A$ and self-adjoint $H_B$ . This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already in Ibort et al. [B...
Article
The spectral geometry of infinite graphs deals with three major themes and their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the heat flow with its probabilistic aspects. In this book, all three themes are brought together coherently under the perspective of Dirichlet forms, providing a powerful and un...
Article
Full-text available
Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods f...
Article
Full-text available
We study domination of quadratic forms in the abstract setting of ordered Hilbert spaces. Our main result gives a characterization in terms of the associated forms. This generalizes and unifies various earlier works. Along the way we present several examples.
Chapter
Analysis and Geometry on Graphs and Manifolds - edited by Matthias Keller August 2020
Article
Cambridge Core - Abstract Analysis - Analysis and Geometry on Graphs and Manifolds - edited by Matthias Keller
Article
Full-text available
We consider topological dynamical systems over \({\mathbb {Z}}\) and, more generally, locally compact, \(\sigma \)-compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of diffraction theory to associate an autocorrelation and a diffraction measure to any \(L^2\)-function ove...
Preprint
Full-text available
We consider metrizable ergodic topological dynamical systems over locally compact, $\sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and sh...
Preprint
Full-text available
We show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure solves what we call the phase problem if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure solves the phase problem independent of...
Preprint
Full-text available
We compute the deficiency spaces of operators of the form $H_A{\hat{\otimes}} I + I{\hat{\otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already by Ibort, Marmo and P\'e...
Preprint
Full-text available
We show that a generic quasi-periodic Schr\"odinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schr\"odinger operator with the resulting potential has...
Preprint
Full-text available
Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyse these structures using methods f...
Preprint
Full-text available
We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple...
Preprint
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet Laplacian on subsets that satis...
Preprint
We study mean equicontinuous actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and provide a characterization of mean equicontinuity of an action via properties of its product. This charact...
Preprint
We consider topological dynamical systems over $\ZZ$ and, more generally, locally compact, $\sigma$-compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of diffraction theory to associate an autocorrelation and a diffraction measure to any $L^2$-function over such a dynamica...
Article
Full-text available
We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk's group acting on the boundary of the infinite binary tree. We establish a connection between the underlying dynamical system and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger op...
Article
We study topological Poincar\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a pr...
Article
Full-text available
We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any intertwining order isomorphism is necessarily unita...
Article
Full-text available
It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we gene...
Article
Full-text available
We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.
Article
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We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green's formula for functions from suitable function spaces and suitable operators arising from extensions of the underlying form. We first...
Article
Full-text available
We consider graphs associated to Delone sets in Euclidean space. Such graphs arise in various ways from tilings. Here, we provide a unified framework. In this context, we study the associated Laplace operators and show Gaussian heat kernel bounds for their semigroups. These results apply to both metric and discrete graphs.
Article
Full-text available
We present a decomposition principle for general regular Dirichlet forms satisfying a spatial local compactness condition. We use the decomposition principle to derive a Persson type theorem for the corresponding Dirichlet forms. In particular our setting covers Laplace-Beltrami operators on Riemannian manifolds, and Dirichlet forms associated to $...
Article
Full-text available
We study combinatorial properties of the subshift induced by the substitution that describes Lysenok’s presentation of Grigorchuk’s group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on the one hand, it allowed embedding Grigorchuk’s group in a topological full group, and on the o...
Article
Full-text available
The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(\alpha^n x)^{}_{n\in\mathbb{N}}$, where $\alpha$ is a fixed real number with $| \alpha | > 1$ and $x\in\mathbb{R}$ is arbitrary. Such sequences appear in a multitude of situations including the spectral theor...
Chapter
This short chapter may be viewed as a complement to the chapters on almost periodicity. Its goal is a fairly self-contained account of some averaging processes of functions along sequences of the form, where α is a fixed real number with and is arbitrary. Such sequences appear in the spectral theory of inflation systems in various ways. Due to the...
Article
Full-text available
We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull i...
Article
Full-text available
In this article, we present a new method to treat uniqueness of form extensions in a rather general setting including various magnetic Schr\"odinger forms. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. We review this concept in an abstract setting and give a characterization in terms of the...
Article
Full-text available
We study dynamical systems $(X,G,m)$ with a compact metric space $X$ , a locally compact, $\unicode[STIX]{x1D70E}$ -compact, abelian group $G$ and an invariant Borel probability measure $m$ on $X$ . We show that such a system has a discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this w...
Article
Full-text available
We prove a quantitative uncertainty principle at low energies for the Laplacian on fairly general weighted graphs with a uniform explicit control of the constants in terms of geometric quantities. A major step consists in establishing lower bounds for Dirichlet eigenvalues in terms of the geometry.
Article
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We construct model sets arising from cut and project schemes in Euclidean spaces whose associated Delone dynamical systems have positive toplogical entropy. The construction works both with windows that are proper and with windows that have empty interior. In a probabilistic construction, the entropy almost surely turns out to be proportional to th...
Article
Full-text available
The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly in...
Article
Full-text available
Various spectral notions have been employed to grasp the structure of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and give generalisations in the form of further...
Article
Full-text available
We show that binary Toeplitz flows can be interpreted as Delone dynamical systems induced by model sets and analyse the quantitative relations between the respective system parameters. This has a number of immediate consequences for the theory of model sets. In particular, we use our results in combination with special examples of irregular Toeplit...
Article
Full-text available
There is a recently discovered connection between the field of Schroedinger operators associated to aperiodic order and the field of Laplacians on Schreier graphs arising from Grigorchuk's group. We give an overview of corresponding results. This includes results on Cantor spectrum of Lebesgue measure zero as well as on absence of eigenvalues for L...
Article
Full-text available
We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are g...
Article
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We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other using explicit expressions. In particular, we show that the averaging operator is closely related with...
Article
Full-text available
The p-Laplacian operators have a rich analytical theory and in the last few years they have also offered efficient tools to tackle several tasks in machine learning. During the workshop mathematicians and theoretical computer scientists working on models based on p-Laplacians on graphs and manifolds have presented the latest theoretical development...
Book
What is order that is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the – later Nobel prize-winning – discovery of quasicrystals, the investigation of aperiodic order ha...
Article
Full-text available
We consider diffraction of Delone sets in Euclidean space. We show that the set of Bragg peaks with high intensity is always Meyer (if it is relatively dense). We use this to provide a new characterization for Meyer sets in terms of positive and positive definite measures. Our results are based on a careful study of positive definite measures, whic...
Article
We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intr...
Article
Full-text available
We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a...
Article
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We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a...
Article
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We investigate (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include periodic graphs, fractal graphs and graphings. Making use of Connes' non-commutative integration theory we construct a Zeta function and present a determinant formula for it. We show that our construction is...
Article
Full-text available
We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. We consider in particular, the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.
Article
We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self-adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self-adjoint operators. We then characterize the lower bounded self-adjoint Laplacians and determine...
Article
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We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown b...
Article
We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown b...
Article
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Trees of finite cone type have appeared in various contexts. In particular, they come up as simplified models of regular tessellations of the hyperbolic plane. The spectral theory of the associated Laplacians can thus be seen as induced by geometry. Here we give an introduction focusing on background and then turn to recent results for (random) per...
Article
We consider minimal, aperiodic symbolic subshifts and show how to characterize the combinatorial property of bounded powers by means of a metric property. For this purpose we construct a family of graphs which all approximate the subshift space, and define a metric on each graph which extends to a metric on the subshift space. The characterization...
Article
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We consider an arbitrary selfadjoint operator on a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators...
Article
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We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure an...
Article
We study Schr\"odinger 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...
Article
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It is well-known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the sp...
Article
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C*-algebras and aperiodic tilings. Our startin...
Article
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We develop the theory of distributive inverse semigroups as the analogue of distributive lattices without top element and prove that they are in a duality with those etale groupoids having a spectral space of identities, where our spectral spaces are not necessarily compact. We prove that Boolean inverse semigroups can be characterized as those dis...
Article
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We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stabili...
Article
In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distri...
Article
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We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all $\ell^p$, $1\leq p < \infty$, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding...
Article
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We consider Delone sets with finite local complexity. We characterize validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In this context, we show that linear repetitivity is equivalent to positivity of weights combined with a certain bala...
Article
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We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact Abelian groups, which provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular we can construct complete families of solutions...
Article
We study repetitions in infinite words coding exchange of three intervals with permutation (3, 2, 1), called 3iet words. The language of such words is determined by two parameters, ϵ,ℓ. We show that finiteness of the index of 3iet words is equivalent to boundedness of the coefficients of the continued fraction of ϵ. In this case, we also give an up...
Article
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We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with nite local complexity the only equicontinuous systems are then shown to be the crystalline ones. On the other hand, within the class without nite local complexity, we exhibit examples o...