Multifractal analysis
Description
Multifractal analysis is, generally speaking, the study of the regularity properties of a locally rough function. Such objects typically exhibit very rich local behaviour: for many of the well-studied examples, the level sets of the functions are dense in their support and are often very large (in the sense of Hausdorff dimension) for a continuum of values.
Much of my work has focused on the geometry of overlapping self-similar iterated function systems for the “hard” range of q<0: see for example my WSC paper or my multifractal decomposition paper. I have also worked on the multifractal analysis of random substitutions.
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Relevant Research Articles
- Assouad spectrum of Gatzouras–Lalley carpets
With: Amlan Banaji, Jonathan M. Fraser, István Kolossváry Journal: Preprint (submitted) Links: pdfarxiv - Multifractal analysis of measures arising from random substitutions
With: Andrew Mitchell Journal: Comm. Math. Phys. 405 (2024), Paper No. 63, 44 p. Links: pdfdoizblarxiv - A multifractal decomposition for self-similar measures with exact overlaps
Journal: Preprint (submitted) Links: pdfarxiv - Local dimensions of self-similar measures satisfying the Finite Neighbour Condition
With: Kathryn E. Hare Journal: Nonlinearity 35 (2022), 4876–4904 Links: pdfdoizblarxiv - Geometric and combinatorial properties of self-similar multifractal measures
Journal: Ergodic Theory Dyn. Syst. 43 (2023), 2028–2072 Links: pdfdoizblarxiv