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Franz Gaehler

Publications and source records attributed to Franz Gaehler.

6 recordsLinked to original sources
arXiv2019

Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction

The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. We also discuss some consequences for the spectral measures of such systems. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations.

math.DS↗
arXiv2015

Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples

This article presents, in an illustrative fashion, a first step towards an extension of the spectral theory of constant length substitutions. Our starting point is the general observation that the symbolic picture (as defined by the substitution rule) and its geometric counterpart with natural prototile sizes (as defined by the induced inflation rule) may differ considerably. On the geometric side, an aperiodic inflation system possesses a set of exact renormalisation relations for its pair correlation coefficients. Here, we derive these relations for some paradigmatic examples and infer various spectral consequences. In particular, we consider the Fibonacci chain, revisit the Thue--Morse and the Rudin--Shapiro sytem, and finally analyse a twisted extension of the silver mean chain with mixed singular spectrum.

math.MG↗
arXiv2014

Determining pure discrete spectrum for some self-affine tilings

By the algorithm implemented in the paper [2] by Akiyama-Lee and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to $2$, and some cases of planar tilings generated by boundary substitutions due to the paper [17] by Kenyon.

math.MG↗
arXiv2012

Integral cohomology of rational projection method patterns

We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomological invariants. Our framework unifies the several previous methods used to study the cohomology of these patterns. We discuss explicit calculations for the main examples of icosahedral patterns in $R^3$ -- the Danzer tiling, the Ammann-Kramer tiling and the Canonical and Dual Canonical $D_6$ tilings, including complete computations for the first of these, as well as results for many of the better known 2 dimensional examples.

math.KT↗
arXiv1996

Atomic Self-Diffusion in Quasicrystals: A Molecular Dynamics Study

We present a molecular dynamics study on atomic self-diffusion in Frank-Kasper type dodecagonal quasicrystals. It is found that the quasicrystal-specific flip mechanism for atomic diffusion as predicted by Kalugin and Katz, indeed occurs in this system. However, in order to be effective, this mechanism needs to be catalyzed by other defects such as half-vacancies if the structure is truly three-dimensional. For this reason, flip diffusion is difficult to distinguish from standard vacancy diffusion. In a quasi-two-dimensional setup, however, the flips may occur without other defects. Activation energies and flip frequencies are also determined.

cond-mat↗