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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• Research Articles
• Expository Articles

Relevant Research Articles

1. Assouad spectrum of Gatzouras–Lalley carpets

   With:	Amlan Banaji, Jonathan M. Fraser, István Kolossváry
   Journal:	Preprint (submitted)
   Links:	pdfarxiv

2. Multifractal analysis of measures arising from random substitutions

   With:	Andrew Mitchell
   Journal:	Comm. Math. Phys. 405 (2024), Paper No. 63, 44 p.
   Links:	pdfdoizblarxiv

3. A multifractal decomposition for self-similar measures with exact overlaps

   Journal:	Preprint (submitted)
   Links:	pdfarxiv

4. Local dimensions of self-similar measures satisfying the Finite Neighbour Condition

   With:	Kathryn E. Hare
   Journal:	Nonlinearity 35 (2022), 4876–4904
   Links:	pdfdoizblarxiv

5. Geometric and combinatorial properties of self-similar multifractal measures

   Journal:	Ergodic Theory Dyn. Syst. 43 (2023), 2028–2072
   Links:	pdfdoizblarxiv

Relevant Expository Articles

1. Multifractal analysis via Lagrange duality

   Journal:	Preprint (submitted)
   Links:	pdfarxiv