Fractal and Statistical Phenomena in Aperiodic Order

In this work, we study (primitive) inflation systems arising from substitutions and tiling rules, in particular their statistical properties and several fractal phenomena appearing while describing them. <br /><br /> We provide a careful analysis of one-dimensional unimodular Pisot substitutions and show a dichotomy result for their corresponding windows in model set description. The resulting set can either be an interval (or a finite union of them) or a Cantroval. <br /><br /> Further, we explore various ways of computing patch frequencies. For dualisation tilings, we describe an algorithm which allows the exact computation of frequencies of patches with an arbitrary number of tiles. <br /><br /> Next, we employ some geometric and combinatorial techniques and extend the known results about pair correlations to higher-order correlations. By that, we derive a general framework for exact computation of frequencies in any primitive inflation system. We also show how to pass from the geometrical point of view to a symbolic one. <br /><br /> Lastly, we merge these two branches together to obtain diffraction patterns of the recently discovered aperiodic monotile tilings, whose windows possess fractal boundaries and the usual techniques fail in their cases. To do so, we provide an explicit formula for the diffraction exact intensities of deformed model sets using the undeformed case.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2025

Author dc:creator

• Mazáč, Jan

Repository record source_url

https://pub.uni-bielefeld.de/record/3006299

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:3006299