arXiv subjects
Publications and source records attributed to Daniel Lenz.
Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.
We prove a variant of Sch'nol's theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with $δ$- or Kirchhoff boundary conditions.
We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS'. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous "periodic" IDS which is regularized by a random perturbation.
It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $β$ and then relate the properties of $β$ to ones of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.
We provide an ergodic theorem for certain Banach-space valued functions on structures over $\ZZ^d$, which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated finite-range operators in the sense of convergence of the distributions with respect to the supremum norm. These results apply to various examples including periodic operators, percolation models and nearest-neighbour hopping on the set of visible points. Our method gives explicit bounds on the speed of convergence in terms of the speed of convergence of the underlying frequencies. It uses neither von Neumann algebras nor a framework of random operators on a probability space.
We present a construction, which assigns two groupoids, $\Gugamma$ and $\Gmgamma$, to an inverse semigroup $Γ$. By definition, $\Gmgamma$ is a subgroupoid (even a reduction) of $\Gugamma$. The construction unifies known constructions for groupoids. More precisely, the groupoid $\Gugamma$ is shown to be isomorphic to the universal groupoid of $Γ$ introduced by Paterson. For $Γ$ arising from graphs resp. tilings, the groupoid $\Gmgamma$ is the graph groupoid introduced by Kumjian et al. resp. the tiling groupoid introduced by Kellendonk. We obtain a characterisation of open invariant sets in $\Gmgamma^{(0)}$ in terms of certain order ideals of $\Gammanull$ for a large class of $Γ$ (including those arising from graphs and from tilings). If $\Gmgamma$ is essentially principal this gives a characterization of the ideal structure of $\Cred(\Gmgamma)$ by a theory of Renault. In particular, we then obtain necessary and sufficient conditions on $Γ$ for simplicity of $\Cred(\Gmgamma)$. Our approach relies on a detailed analysis of the order structure of $Γ$.
We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of factor maps between the case of continuous and non-continuous weight functions.
We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this for a large class of subshifts, namely those satisfying Boshernitzan's condition. This is accomplished by relating the essential support to uniform convergence properties of the corresponding Szego cocycles.
This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under ``equivariant'' local perturbations and discuss various examples,including deformed model sets. A key step in the proof of stability consists in transforming the problem into a question on factors of dynamical systems.
We prove quantum dynamical lower bounds for one-dimensional continuum Schrödinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including the Bernoulli-Anderson model with a constant single-site potential.
We show that geometric disorder leads to purely singular continuous spectrum generically. The main input is a result of Simon known as the ``Wonderland theorem''. Here, we provide an alternative approach and actually a slight strengthening by showing that various sets of measures defined by regularity properties are generic in the set of all measures on a locally compact metric space. As a byproduct we obtain that a generic measure on euclidean space is singular continuous.
Certain topological dynamical systems are considered that arise from actions of $σ$-compact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan, every locally constant SL(2,R)-valued cocycle is uniform. As a consequence, the corresponding Schrödinger operators exhibit Cantor spectrum of Lebesgue measure zero. An investigation of Boshernitzan's condition then shows that these results cover all earlier results of this type and, moreover, provide various new ones. In particular, Boshernitzan's condition is shown to hold for almost all circle maps and almost all Arnoux-Rauzy subshifts.
We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of the integrated density of states in the sense of uniform convergence in distribution for the associated random operators.
We consider a random family of Schrödinger operators on a cover $X$ of a compact Riemannian manifold $M = X/Γ$. We present several results on their spectral theory, in particular almost sure constancy of the spectral components and existence and non-randomness of an integrated density of states. We also sketch a groupoid based general framework which allows to treat basic features of random operators in different contexts in a unified way. Further topics of research are also discussed.
We study ergodic random Schr"odinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a selfaveraging integrated density of states and a Šubin type trace formula.
We show that aperiodic linearly repetitive Delone sets are densely repetitive. This confirms a conjecture of Lagarias and Pleasants.
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.