Michael Baake

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

Certain topological dynamical systems are considered that arise from actions of

PLANAR PATTERNS WITH FIVEFOLD SYMMETRY AS SECTIONS OF PERIODIC STRUCTURES IN 4-SPACE

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A Guide to Mathematical Quasicrystals

-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.

Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability

Two quasiperiodic planar patterns with fivefold symmetry are derived from the root lattice A4 in 4-space. A detailed analysis of the geometry of the A4 Voronoi complex and its dual complex is presented with special emphasis on fivefold symmetry. By means of the general dualization method, 2D patterns are obtained, one with triangular tiles and a second which turns out to be the well-known Penrose pattern. The vertex configurations and their relative frequencies, the deflation rules, and the Fourier properties of these patterns are worked out in the framework of the dualization method and Klotz construction.

Characterization of model sets by dynamical systems

This contribution deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular, we focus on their interplay with various physically motivated equivalence concepts such as local indistinguishability and local equivalence. Various discrete patterns with non-crystallographic symmetries are described in detail, and some of their magic properties are introduced. This perfectly ordered world is augmented by a brief introduction to the stochastic world of random tilings.

Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces

Two 2D quasiperiodic tilings with generalized tenfold symmetry are derived from the lattice A4R, the reciprocal of the root lattice A4. Both tilings are built from four tiles, triangles in one case, rhombi and hexagons in the other. After a brief description of the tilings and their structures, the authors introduce the equivalence concept of mutual local derivability. They present its key properties and its application to several tenfold tilings and discuss some implications on a future classification of tilings in position space.

Trace maps as 3D reversible dynamical systems with an invariant

It is shown how regular model sets can be characterized in terms of the 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

TRACE MAPS, INVARIANTS, AND SOME OF THEIR APPLICATIONS

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Diffraction of Random Tilings: Some Rigorous Results

and then relate the properties of

The torus parametrization of quasiperiodic LI-classes

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