Multifractal analysis for g-measures and singular potentials
Final Report Abstract
From the structure of snowflakes and Romanesco broccoli to the patterns of earthquakes and the coastline of Britain - fractals can be found everywhere in nature. In science, they keep reappearing in many fields such as Biology, Medicine, Physics, Chemistry and Mathematics. Fractals are characterised by their self-similarity and an intricate local structure that is observable at a wide range of resolutions. In several mathematical areas like number theory or the study of chaotic dynamical systems, the result of painting all points with the same behaviour with the same colour often leads to a picture with fractal features. In fact, portraying all colours at once gives a whole collection of fractals, called a multifractal. This happens in particular, if we keep sampling a given observable (e.g. the temperature) of a dynamical system and colour the initial state according to the mean value. In this project, we study the corresponding multifractals in a setting where the observable can take arbitrarily large values, extending the already existing tools and techniques and pointing out new phenomena. In the second part of the project, we use this enhanced toolbox in fractal geometry to analyse and categorise self-similar structures in the fields of aperiodic order and number theory. In particular, we gain a deeper understanding of paradigmatic sequences like the Thue–Morse sequence and the class of regular sequences. Our approach also extends to mathematical models for quasicrystals, the famous Penrose tiling and the recently discovered hat tiling that has gained widespread popularity for providing an aperiodic monotile. We also make progress on the class of random substitutions, a probabilistic framework that creates fractals with a higher complexity than classical constructions.
Publications
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Spectral theory of regular sequences: parametrisation and spectral characterisation.
M. Coons, J. Evans, P. Gohlke & N. Maibo
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A classification of intrinsic ergodicity for recognisable random substitution systems
P. Gohlke & A. Mitchell
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Fast Dimension Spectrum for a Potential with a Logarithmic Singularity. Journal of Statistical Physics, 191(3).
Gohlke, Philipp; Lamprinakis, Georgios & Schmeling, Jörg
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Orbit separation dimension as complexity measure for primitive inflation tilings. Ergodic Theory and Dynamical Systems, 45(10), 2992-3020.
BAAKE, MICHAEL; GÄHLER, FRANZ & GOHLKE, PHILIPP
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Rauzy fractals of random substitutions. Advances in Mathematics, 485, 110713.
Gohlke, P.; Mitchell, A.; Rust, D. & Samuel, T.
