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Multifractal analysis for the pointwise Assouad dimension of self-similar measures

Roope Anttila, Ville Suomala

TL;DR

This work develops a multifractal analysis for the pointwise Assouad dimension of self-similar measures under the open set condition (OSC). It first clarifies that global doubling is too coarse a notion, showing that such measures are pointwise doubling on a set of full Hausdorff dimension while non-doubling occurs on sets of full measure, and then determines the multifractal spectrum of the level sets of the pointwise Assouad dimension. The authors introduce a symbolic, large-deviation based approach (method of types) and construct large Moran subsets to prove that, for Bernoulli measures, the Hausdorff dimension of the α-level sets D^A_α coincides with the upper spectrum ḟ(α) for α in [α_min, α_max], with precise behavior at the endpoints and an explicit transfer to self-similar measures under OSC. The results extend to non-doubling self-similar measures, showing that D(μ) can have full dimension but zero s-dimensional measure, and they establish a broad framework for the pointwise doubling properties of self-similar structures, bridging symbolic dynamics and geometric measure theory.

Abstract

We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full Hausdorff dimension, despite the fact that they can in general be non-doubling in a set of full Hausdorff measure. More generally, we carry out multifractal analysis by determining the Hausdorff dimension of the level sets of the pointwise Assouad dimension.

Multifractal analysis for the pointwise Assouad dimension of self-similar measures

TL;DR

This work develops a multifractal analysis for the pointwise Assouad dimension of self-similar measures under the open set condition (OSC). It first clarifies that global doubling is too coarse a notion, showing that such measures are pointwise doubling on a set of full Hausdorff dimension while non-doubling occurs on sets of full measure, and then determines the multifractal spectrum of the level sets of the pointwise Assouad dimension. The authors introduce a symbolic, large-deviation based approach (method of types) and construct large Moran subsets to prove that, for Bernoulli measures, the Hausdorff dimension of the α-level sets D^A_α coincides with the upper spectrum ḟ(α) for α in [α_min, α_max], with precise behavior at the endpoints and an explicit transfer to self-similar measures under OSC. The results extend to non-doubling self-similar measures, showing that D(μ) can have full dimension but zero s-dimensional measure, and they establish a broad framework for the pointwise doubling properties of self-similar structures, bridging symbolic dynamics and geometric measure theory.

Abstract

We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full Hausdorff dimension, despite the fact that they can in general be non-doubling in a set of full Hausdorff measure. More generally, we carry out multifractal analysis by determining the Hausdorff dimension of the level sets of the pointwise Assouad dimension.
Paper Structure (10 sections, 21 theorems, 109 equations, 1 figure)

This paper contains 10 sections, 21 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Self-similar sets and measures
  4. Symbolic spaces and Bernoulli measures
  5. Multifractal analysis of pointwise dimensions
  6. Multifractal analysis of Bernoulli measures
  7. Method of types
  8. Multifractal analysis
  9. Construction of large Moran subsets
  10. Multifractal analysis of self-similar measures

Key Result

Theorem 1.1

Let $\mu$ be a non-doubling self-similar measure fully supported on a self-similar set $X\subset\mathbb{R}^n$ satisfying the OSC and let $s=\mathop{\mathrm{dim_H}}\nolimits X$. Then

Figures (1)

  • Figure 1: Proof of Lemma \ref{['lemma:case-study']}: The point in the picture is $x\in A_2$ and letting $d=d(x,\varphi_{\mathtt{i}_2}(X))$, we see that $B(x,2d)$ contains a sub-cylinder of $[\mathtt{i}_2]$ of size comparable to $d$. This gives a lower bound for $\mu(B(x,d))$. The choice of $N_2$ ensures that a sufficiently large portion of the mass of $B(x,R)$ comes from $[\mathtt{i}_2]$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 29 more