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Daniel Lenz

Publications and source records attributed to Daniel Lenz.

At least 37 records · Page 2Linked to original sources
arXiv2020

Modulated crystals and almost periodic measures

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 from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similarly to de Bruijn's analysis, we find stability with respect to almost periodic modulations.

math.DS↗
arXiv2020

Uniqueness of form extensions and domination of semigroups

In this article, we present a new method to study uniqueness of form extensions in a rather general setting. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. Our main abstract result transfers uniqueness of form extension of a dominating form to that of a dominated form. This result can be applied to a multitude of examples including various magnetic Schrödinger forms on graphs and on manifolds.

math.FA↗
arXiv2020

Pure point spectrum for dynamical systems and mean almost periodicity

We consider metrizable ergodic topological dynamical systems over locally compact, $σ$-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 show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.

math.DS↗
arXiv2019

The Nature of Ionized Gas in the Milky Way Galactic Fountain

We address the spatial scale, ionization structure, mass and metal content of gas at the Milky Way disk-halo interface detected as absorption in the foreground of seven closely-spaced, high-latitude halo blue horizontal branch stars (BHBs) with heights z = 3 - 14 kpc. We detect transitions that trace multiple ionization states (e.g. CaII, FeII, SiIV, CIV) with column densities that remain constant with height from the disk, indicating that the gas most likely lies within z < 3.4 kpc. The intermediate ionization state gas traced by CIV and SiIV is strongly correlated over the full range of transverse separations probed by our sightlines, indicating large, coherent structures greater than 1 kpc in size. The low ionization state material traced by CaII and FeII does not exhibit a correlation with either N$_{\rm HI}$ or transverse separation, implying cloudlets or clumpiness on scales less than 10 pc. We find that the observed ratio log(N_SiIV/ N_CIV), with a median value of -0.69+/-0.04, is sensitive to the total carbon content of the ionized gas under the assumption of either photoionization or collisional ionization. The only self-consistent solution for photoionized gas requires that Si be depleted onto dust by 0.35 dex relative to the solar Si/C ratio, similar to the level of Si depletion in DLAs and in the Milky Way ISM. The allowed range of values for the areal mass infall rate of warm, ionized gas at the disk-halo interface is 0.0003 < dM_gas / dtdA [M_sun kpc^-2 yr^-] < 0.006. Our data support a physical scenario in which the Milky Way is fed by complex, multiphase processes at its disk-halo interface that involve kpc-scale ionized envelopes or streams containing pc-scale, cool clumps.

astro-ph.GA↗
arXiv2019

The structure of mean equicontinuous group actions

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 characterization enables us to show the equivalence of mean equicontinuity and the weaker notion of Besicovitch-mean equicontinuity in fairly high generality, including actions of abelian groups as well as minimal actions of general groups. In the minimal case, we further conclude that mean equicontinuity is equivalent to discrete spectrum with continuous eigenfunctions. Applications of our results yield a new class of non-abelian mean equicontinuous examples as well as a characterization of those extensions of mean equicontinuous actions which are still mean equicontinuous.

math.DS↗
arXiv2019

Large-scale Maps of the Cosmic Infrared Background from Planck

The cosmic infrared background (CIB) is a powerful probe of large-scale structure across a very large redshift range, and consists of unresolved redshifted infrared emission from dusty galaxies. It can be used to study the astrophysics of galaxies, the star formation history of the universe, and the connection between dark and luminous matter. It can furthermore be used as a tracer of the large-scale structure and thus assist in de-lensing of the cosmic microwave background. The major difficulty in its use lies in obtaining accurate and unbiased large-scale CIB images that are cleaned of the contamination by Galactic dust. We used data on neutral atomic hydrogen from the recently-released HI4PI Survey to create template maps of Galactic dust, allowing us to remove this component from the Planck intensity maps from 353 to 857 GHz for approximately $25\%$ of the sky. This allows us to constrain the CIB power spectrum down to $\ell\gtrsim 70$. We present these CIB maps and the various processing and validation steps that we have performed to ensure their quality, as well as a comparison with previous studies. All our data products are made publicly available at https://doi.org/10.7910/DVN/8A1SR3, thereby enabling the community to investigate a wide range of questions related to the universe's large-scale structure.

astro-ph.CO↗
arXiv2019

Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schrödinger Operators on the Real Line

We show that a generic quasi-periodic Schrödinger 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 Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.

math.SP↗
arXiv2019

Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

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 Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

math.DS↗
arXiv2019

Universal lower bounds for Laplacians on weighted graphs

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 satisfy natural geometric conditions.

math.DG↗
arXiv2018

Spectral theory of dynamical systems as diffraction theory of sampling functions

We consider topological dynamical systems over $\ZZ$ and, more generally, locally compact, $σ$-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 dynamical system. This diffraction measure is shown to be the spectral measure of the function. If the group has a countable basis of the topology one can also exhibit the underlying autocorrelation by sampling along the orbits. Building on these considerations we then show how the spectral theory of dynamical systems can be reformulated via diffraction theory of function dynamical systems. In particular, we show that the diffraction measures of suitable factors provide a complete spectral invariant.

math.DS↗
arXiv2018

Model sets with positive entropy in Euclidean cut and project schemes

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 the measure of the boundary of the window.

math.DS↗
arXiv2018

Geometric properties of Dirichlet forms under order isomorphisms

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 unitary (up to a constant). In the topological setting we deal with quasi-regular forms and show that any intertwining order isomorphism induces a quasi-homeomorphism between the underlying spaces. In the geometric setting we deal with both regular Dirichlet forms as well as resistance forms and essentially show that the geometry defined by these forms is preserved by intertwining order isomorphisms. In particular, we prove in the strongly local regular case that intertwining order isomorphisms induce isometries with respect to the intrinsic metrics between the underlying spaces under fairly mild assumptions. This applies to a wide variety of metric measure spaces including $\mathrm{RCD}(K,N)$-spaces, complete weighted Riemannian manifolds and complete quantum graphs. In the non-local regular case our results cover in particular graphs as well as fractional Laplacians as arising in the treatment of $α$-stable Lévy processes. For resistance forms we show that intertwining order isomorphisms are isometries with respect to the resistance metrics. Our results can can be understood as saying that diffusion always determines the Hilbert space, and -- under natural compatibility assumptions -- the topology and the geometry respectively. As special instances they cover earlier results for manifolds and graphs.

math.FA↗
arXiv2018

The Ihara Zeta function for infinite graphs

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and percolation graphs. Making use of Connes' non-commutative integration theory we construct a Zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara Zeta function via the normalized version on finite graphs in the sense of Benjamini-Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs.

math.MG↗
arXiv2017

Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order

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 action of $G$ on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain specific graphs. The methods developed here apply to a large class of examples.

math.GR↗
arXiv2017

Combinatorics of the subshift associated to Grigorchuk's group

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 one hand, it allowed to embed Grigorchuk's group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs.

math.DS↗
arXiv2017

Domination of quadratic forms

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.

math.FA↗