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.
- Multifractal analysis of measures arising from random substitutions
With: Andrew Mitchell Journal: Preprint (submitted) Links: pdfarxiv
- 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