Hausdorff Measure and Dimension with Examples

TL;DR

This work presents two principal approaches to computing the Hausdorff dimension: the Mass Distribution Principle-based lower/upper bound method and the Similarity Dimension method for self-similar sets. It provides explicit, step-by-step demonstrations on the unit square and the Cantor set, illustrating how coverings, mass distributions, and iterated function system (IFS) structures yield exact dimensions such as $\dim_{\mathcal{H}}(S)=2$ and $\dim_{\mathcal{H}}(C)=\frac{\log 2}{\log 3}$, with the Moran-Hutchinson Open Set Condition guaranteeing equality between the similarity and Hausdorff dimensions. The paper clarifies theoretical underpinnings like invariance of $\dim_{\mathcal{H}}$ under covering shapes and the use of the mass distribution principle to obtain nontrivial lower bounds. By coupling rigorous derivations with classical self-similar examples, it highlights practical pathways for fractal-dimension computation in metric spaces. The results reinforce the relevance of OSC and IFS-based approaches for accurate dimension assessment in fractal geometry.

Abstract

This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for calculating the dimension of a set, ensuring all calculation details are transparent for the reader. The paper first establishes the theoretical groundwork by reviewing the definitions of the Hausdorff measure, proving the dimensional invariance under changes to the shape of the covering sets, and confirming the dimensional property of open sets. It then introduces the two main methodologies. The first is the Lower and Upper Bound Estimation, which uses the relationship between the measure $H^s(A)$ and the dimension $\dim_{H}(A)$. This method emphasizes the use of the Mass Distribution Principle for establishing the lower bound, which is essential when the Lebesgue measure of the set is zero. The second, more computationally efficient method is the Similarity Dimension Method, introduced via the definitions of Similitudes, Iterated Function Systems (IFS), and the Moran-Hutchinson Theorem. Both methodologies are applied rigorously to classic examples: the Unit Square (yielding dimension $2$) and the Cantor Set (yielding the fractional dimension $\log 2 / \log 3$). The paper serves as a detailed, step-by-step guide intended to make the application of these fundamental fractal geometry concepts clearer and easier to understand.

Figures (2)

• Figure 1: The Phase Transition of the Hausdorff Measure $\mathcal{H}^s(A)$. The Hausdorff Dimension is the critical exponent ($s$) where the measure drops from infinity to zero.
• Figure 2: The construction of the Middle-Thirds Cantor Set (C) over six stages (n=0 to n=5). At each stage, the open middle third is removed from every remaining interval. This process highlights the self-similar nature of the set, where the initial interval is scaled down by a factor r=1/3 and copied N=2 times at each iteration. This scaling relationship (N=2,r=1/3) is key to calculating the Hausdorff dimension using the Similarity Dimension method, yielding $\dim_\mathcal{H}(C)=\log 2/\log 3$ (see Section \ref{['sec:sim-meth']}).

Theorems & Definitions (21)

• Definition 2.1: Spherical Hausdorff Measure
• Definition 2.2: Hausdorff Definition (Generic Sets)
• Theorem 2.1: $\mathcal{H}_\delta^s$ is a measure on $\mathbb{R}^n$
• Theorem 2.2: $\mathcal{H}_\delta^s$ is Borel Regular Measure on $\mathbb{R}^n$
• Definition 2.3: $s$-dimensional Hausdorff measure
• Lemma 2.3: Properties of $\mathcal{H}^s$
• Definition 2.4: Hausdorff Dimension
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6: Invariance of Hausdorff Dimension from the Shape of Sets