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Hausdorff measure of dominated planar self-affine sets with large dimension

Balázs Bárány

TL;DR

The paper analyzes planar dominated self-affine sets at their affinity dimension, establishing that a positive and finite $s_0$-dimensional Hausdorff measure ${\cal H}^{s_0}(X)$ is equivalent to the Käenmäki measure being a mass distribution, and, under the open bounded neighbourhood condition, to the projection of this measure along every Furstenberg direction having a bounded density. It leverages a Perron-Frobenius framework to connect slice-content and projection properties via an eigenfunction $p$ and measure $\nu$, and introduces an alternative form of the Käenmäki measure through $\eta_{\overline{\imath}}$, yielding a dichotomy of trivial vs. equivalent measures. The main theorem ties these structural conditions to the positivity of ${\cal H}^{s_0}(X)$, the boundedness of projected measures, and the equality of the Assouad and affinity dimensions, with consequences for-dimensional regularity under OBNC. The paper then verifies concrete examples, including a diagonal case and a triangular case with Cantor-like Furstenberg directions, using Fourier-analytic methods to establish absolute continuity of projections and thus the required positivity results. These results contribute to a deeper understanding of when self-affine sets exhibit robust, positively measured fractal structure beyond the self-similar setting, and they connect geometric, measure-theoretic, and projection-based criteria in the dominated planar regime.

Abstract

In this paper, we investigate the Hausdorff measure of planar dominated self-affine sets at their affinity dimension. We show that the Hausdorff measure being positive and finite is equivalent to the Käenmäki measure being a mass distribution. Moreover, under the open bounded neighbourhood condition, we will show that the positivity of the Hausdorff measure is equivalent to the projection of the Käenmäki measure in every Furstenberg direction being absolutely continuous with bounded density. This also implies that the affinity and the Assouad dimension coincide. We will also provide examples for both of the cases when the Hausdorff measure is zero and positive.

Hausdorff measure of dominated planar self-affine sets with large dimension

TL;DR

The paper analyzes planar dominated self-affine sets at their affinity dimension, establishing that a positive and finite -dimensional Hausdorff measure is equivalent to the Käenmäki measure being a mass distribution, and, under the open bounded neighbourhood condition, to the projection of this measure along every Furstenberg direction having a bounded density. It leverages a Perron-Frobenius framework to connect slice-content and projection properties via an eigenfunction and measure , and introduces an alternative form of the Käenmäki measure through , yielding a dichotomy of trivial vs. equivalent measures. The main theorem ties these structural conditions to the positivity of , the boundedness of projected measures, and the equality of the Assouad and affinity dimensions, with consequences for-dimensional regularity under OBNC. The paper then verifies concrete examples, including a diagonal case and a triangular case with Cantor-like Furstenberg directions, using Fourier-analytic methods to establish absolute continuity of projections and thus the required positivity results. These results contribute to a deeper understanding of when self-affine sets exhibit robust, positively measured fractal structure beyond the self-similar setting, and they connect geometric, measure-theoretic, and projection-based criteria in the dominated planar regime.

Abstract

In this paper, we investigate the Hausdorff measure of planar dominated self-affine sets at their affinity dimension. We show that the Hausdorff measure being positive and finite is equivalent to the Käenmäki measure being a mass distribution. Moreover, under the open bounded neighbourhood condition, we will show that the positivity of the Hausdorff measure is equivalent to the projection of the Käenmäki measure in every Furstenberg direction being absolutely continuous with bounded density. This also implies that the affinity and the Assouad dimension coincide. We will also provide examples for both of the cases when the Hausdorff measure is zero and positive.

Paper Structure

This paper contains 12 sections, 18 theorems, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. Preliminaries
  5. Perron-Frobenius operator and its eigenfunction
  6. Hausdorff content of slices
  7. Hausdorff measure of slices
  8. An alternative form of the Käenmäki measure
  9. Characterisation of positive measure
  10. Verification of the examples
  11. Diagonal example
  12. Example with positive dimensional Furstenberg directions

Key Result

Theorem 1.1

Let $\Phi=\{f_i(x)=A_ix+t_i\}_{i\in{\cal A}}$ be a dominated planar IFS of affinities with affinity dimension $s_0\in(1,2]$. Let $X$ be the attractor of $\Phi$, let $\mu_K$ be the Käenmäki measure and let $\pi$ be the natural projection. Then the following are equivalent:

Figures (2)

  • Figure 1: First level cylinder sets of the IFS $\Phi=\{f_1(x,y)=\left(\frac{x}{3},\frac{y}{5}\right),f_2(x,y)=\left(\frac{x}{3},\frac{y+2}{5}\right),f_3(x,y)=\left(\frac{x}{3},\frac{y+4}{5}\right),f_4(x,y)=\left(\frac{x+2}{3},\frac{y}{5}\right),f_5(x,y)=\left(\frac{x+2}{3},\frac{y+4}{5}\right),f_6(x,y)=\left(\frac{2x+y+5}{10},\frac{x+2y+2}{10}\right)\}$. Simple calculation shows that the Hausdorff dimension is at most $1.607$ but the largest horizontal slice has dimension $0.6826$, and so, the proper dimensional Hausdorff measure is zero.
  • Figure 2: First level cylinder sets of the IFSs in \ref{['ex:singlexf']} and \ref{['ex:posdimxf']}, which has positive and finite Hausdorff measure.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 29 more