Alex Rutar

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

  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: Permanent Preprint
    Links: pdfarxiv