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Multivariate multifractal analysis of Levy functions. Part I: Determination of multifractal spectra

Stéphane Jaffard, Lingmin Liao, Qian Zhang

TL;DR

This paper conducts a detailed multivariate multifractal analysis for Lévy functions and their translations in a fixed base $b$. By linking Hölder regularity to $b$-adic approximation via the exponent $\Delta^b$, it derives an explicit bivariate spectrum: $\mathcal{D}_{L_{\alpha_1}^b,L_{\alpha_2}^{b,y}}(H_1,H_2)=\min\{H_1/\alpha_1, H_2/\alpha_2\}$ on a region $K_{\alpha_1,\alpha_2}(y)$ determined by $b$-adic approximation, and $-\infty$ outside; for almost every $y$, this reduces to the full rectangle with the min formula. The authors construct Cantor-type subsets, build an appropriate mass distribution, and apply Billingsley’s lemma to obtain the lower bounds, establishing a dichotomy: level sets are either empty or maximal in dimension. They also show that, off the generic region, the spectrum can depend on the explicit $b$-ary expansion of the translation $y$, not just its $b$-adic approximation exponent, highlighting rich subtle dependencies in multivariate spectra. The work lays groundwork for Part II on multivariate formalism validity and suggests extensions to different bases and related jump functions.

Abstract

We study the sets of points where a Lévy function and a translated Lévy function share a given couple of H\''older exponents, and we investigate how their Hausdorff dimensions depend on the translation parameter.

Multivariate multifractal analysis of Levy functions. Part I: Determination of multifractal spectra

TL;DR

This paper conducts a detailed multivariate multifractal analysis for Lévy functions and their translations in a fixed base . By linking Hölder regularity to -adic approximation via the exponent , it derives an explicit bivariate spectrum: on a region determined by -adic approximation, and outside; for almost every , this reduces to the full rectangle with the min formula. The authors construct Cantor-type subsets, build an appropriate mass distribution, and apply Billingsley’s lemma to obtain the lower bounds, establishing a dichotomy: level sets are either empty or maximal in dimension. They also show that, off the generic region, the spectrum can depend on the explicit -ary expansion of the translation , not just its -adic approximation exponent, highlighting rich subtle dependencies in multivariate spectra. The work lays groundwork for Part II on multivariate formalism validity and suggests extensions to different bases and related jump functions.

Abstract

We study the sets of points where a Lévy function and a translated Lévy function share a given couple of H\''older exponents, and we investigate how their Hausdorff dimensions depend on the translation parameter.
Paper Structure (15 sections, 9 theorems, 124 equations, 6 figures)

This paper contains 15 sections, 9 theorems, 124 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Lévy functions
  4. Bivariate multifractal spectrum
  5. Preliminaries
  6. Hölder exponent of Lévy functions and Diophantine approximation
  7. ary expansion
  8. Multivariate multifractal spectra of Lévy functions
  9. Construction of Cantor type subsets
  10. Subsets of Lg
  11. A mass distribution and estimation of the lower local pointwise dimensions
  12. Proof of Theorem \ref{['th']}
  13. Dichotomy of the size of lg
  14. Proof of Propositions \ref{['prop3']} and \ref{['prop1']}
  15. Conclusion

Key Result

Theorem 1.3

Let $O(n)$ denote the orthogonal group of $\mathbb{R}^d$ and $\theta_d$ be its Haar probability measure. Let $\tau_x$ be the translation defined by $\tau_x(y)=x+y$ for any $y\in\mathbb{R}$. For $d\geqslant 2$, and $s,\ t >0$ such that $s+t\geqslant d$, $t>(d+1)/2$, if $A$ is $\mathcal{H}^s$ measurab

Figures (6)

  • Figure 1: The spectrum on the red line is $\frac{H_1}{{\alpha_1}}$, and the spectrum on the blue rectangle is $\min\left\{\frac{H_1}{{\alpha_1}}, \frac{H_2}{{\alpha_2}}\right\}.$
  • Figure 3: The $b$-ary expansion of $y$.
  • Figure 5: The $b$-ary expansion of $y$ (here we take the example of $(l_{i+1}-\lfloor l_i\eta\rfloor-1)/2$ being an even number).
  • Figure 6: The construction of $\mathcal{W}_{2k+1}$.
  • Figure 7: The construction of $\mathcal{W}_{2k+2}$.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Remark 1
  • Theorem 2.5
  • ...and 11 more