Pure Point Dynamical Spectrum Research Articles

AN APERIODIC TILE WITH EDGE-TO-EDGE ORIENTATIONAL MATCHING RULES

AbstractWe present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

Dynamics on the graph of the torus parametrization

Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.

Close-packed Dimers on the Line: Diffraction versus Dynamical Spectrum

The translation action of ℝd on a translation bounded measure ω leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of ω, which is the carrier of the diffraction measure, lives on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) by van Enter and Miȩkisz (J. Stat. Phys. 66:1147–1153, 1992). Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of ‘atomic’ versus ‘molecular’ spectrum suggests a way to come to a more general correspondence between these two types of spectra.

Algorithm for determining pure pointedness of self-affine tilings

Overlap coincidence in a self-affine tiling in R d is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in R d and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several examples. In the course of the proof of the algorithm, we show a variant of the conjecture of Urbański (Solomyak (2006) [40]) on the Hausdorff dimension of the boundaries of fractal tiles.

Pure point spectrum for measure dynamical systems on locally compact Abelian groups

We show equivalence of pure point diffraction and pure point dynamical spectrum for measurable dynamical systems built from locally finite measures on locally compact Abelian groups. This generalizes all earlier results of this type. Our approach is based on a study of almost periodicity in a Hilbert space. It allows us to set up a perturbation theory for arbitrary equivariant measurable perturbations.

Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra

The paper investigates how correlations can completely specify a uniformly discrete point process. The setting is that of uniformly discrete point sets in real space for which the corresponding dynamical hull is ergodic. The first result is that all of the essential physical information in such a system is derivable from its $n$-point correlations, $n= 2, 3, >...$. If the system is pure point diffractive an upper bound on the number of correlations required can be derived from the cycle structure of a graph formed from the dynamical and Bragg spectra. In particular, if the diffraction has no extinctions, then the 2 and 3 point correlations contain all the relevant information.

Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra

Certain topological dynamical systems are considered that arise from actions of $\sigma$-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.

Pure Point Dynamical and Diffraction Spectra

We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.