Multifractal analysis via Lagrange duality

Abstract

We provide a self-contained exposition of the well-known multifractal formalism for self-similar measures satisfying the strong separation condition. At the heart of our method lies a pair of quasiconvex optimization problems which encode the parametric geometry of the Lagrange dual associated with the constrained variational principle. We also give a direct derivation of the Hausdorff dimension of the level sets of the upper and lower local dimensions by exploiting certain weak uniformity properties of the space of Bernoulli measures.

arXiv comment31 pages, 3 figures. Expository article. v2: slightly more general definition of uniform densities; removal of SSC assumption in Proposition 3.13; numbering changes and improvement of exposition; v3: minor fixes. To appear as an expository article in Real Analysis Exchange