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Multifractal analysis via Lagrange duality

Alex Rutar

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.

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.
Paper Structure (18 sections, 20 theorems, 130 equations, 3 figures)

This paper contains 18 sections, 20 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prelude on multifractal analysis
  3. Constrained optimization and duality
  4. Self-similar sets and measures
  5. Shift space and symbolic coding
  6. Dimensions and Lq-spectra of self-similar measures
  7. Hausdorff dimension of self-similar measures
  8. Lq-spectra of self-similar measures
  9. An explicit formula for the Lq-spectrum
  10. Asymptotes of the Lq-spectra
  11. Lagrange duality and multifractal formalism
  12. Continuous optimization and duality
  13. Multifractal spectrum and general upper bound
  14. Multifractal formalism for self-similar measures
  15. Alternative proof of the variational principle
  16. ...and 3 more sections

Key Result

Lemma 2.2

For all sufficiently small $\delta>0$ and for all $\bm{p}\in\mathcal{P}$ with $p_i>0$ for all $i\in\mathcal{I}$, there is a constant $c=c(\bm{p},\delta)>0$ so that for all $\mathtt{i}\in\mathcal{I}^*$ and $x\in S_\mathtt{i}(K)$,

Figures (3)

  • Figure 1: Plot of the $L^q$-spectrum of the measure $\mu_{\bm{p}}$, along with its asymptotes.
  • Figure 2: Plot of the multifractal spectrum $f_{\bm{p}}(\alpha)$.
  • Figure 3: Parametrization of the optimization vector in the simplex $\mathcal{P}\subset\mathop{\mathrm{\mathbb{R}}}\nolimits^3$ and contour lines of $\bm{w}\mapsto q\cdot u(\bm{w})-v(\bm{w})$.

Theorems & Definitions (47)

  • Definition 2.1
  • Lemma 2.2
  • Proof 1
  • Lemma 2.3
  • Proof 2
  • Proposition 2.4
  • Proof 3
  • Lemma 2.5
  • Proof 4
  • Lemma 2.6
  • ...and 37 more