Hausdorff dimension is a powerful tool for measuring the "size" of sets, especially those with irregular shapes. It captures fractional dimensions and provides insights into scaling properties, making it crucial for analyzing complex structures like fractals.

Calculating Hausdorff dimension involves using Hausdorff measure and examining how sets scale. For self-similar sets, there's a simple formula based on the number of copies and scaling factor. This approach reveals fascinating dimensions for well-known fractals like the Cantor set.

Hausdorff dimension and measure

Definition and properties

• Hausdorff dimension is a non-negative real number that provides a way to measure the "size" of a set, taking into account its scaling properties and irregularities
• Defined using Hausdorff measure, which is a generalization of the concept of Lebesgue measure
  • For a set $F \subset \mathbb{R}^n$ and a non-negative real number $s$, the $s$-dimensional Hausdorff measure $H^s(F)$ is defined as the infimum of the sum of the $s$th powers of the diameters of covers of $F$ by sets of diameter at most $\delta$, as $\delta \to 0$
• Hausdorff dimension $\dim_H(F)$ of a set $F$ is the infimum of all $s \geq 0$ such that $H^s(F) = 0$, or equivalently, the supremum of all $s \geq 0$ such that $H^s(F) = \infty$
• Captures the notion of "fractional dimension" and can be non-integer for irregular sets (fractals)
  • Example: The Hausdorff dimension of the Cantor set is $\log(2) / \log(3) \approx 0.631$, which is non-integer

Relation to Hausdorff measure

• Hausdorff dimension is closely related to Hausdorff measure
• For a set $F$, if $s < \dim_H(F)$, then $H^s(F) = \infty$, and if $s > \dim_H(F)$, then $H^s(F) = 0$
• At the critical value $s = \dim_H(F)$, the Hausdorff measure $H^s(F)$ may be zero, infinite, or finite and positive
  • Example: For the Cantor set $C$, $H^s(C) = \infty$ for $s < \log(2) / \log(3)$, $H^s(C) = 0$ for $s > \log(2) / \log(3)$, and $H^s(C) = 1$ for $s = \log(2) / \log(3)$

Hausdorff dimension of self-similar sets

Calculating Hausdorff dimension

• Self-similar sets are sets that exhibit similar patterns at different scales, and their Hausdorff dimension can often be calculated explicitly
• For a self-similar set $F \subset \mathbb{R}^n$ that is the union of $N$ copies of itself, each scaled by a factor of $r$, the Hausdorff dimension $\dim_H(F)$ satisfies the equation $N \times r^{\dim_H(F)} = 1$
  • This equation can be solved for $\dim_H(F)$ using logarithms: $\dim_H(F) = \log(N) / \log(1/r)$

Examples of self-similar sets

• The Cantor set is the union of two copies of itself, each scaled by a factor of $1/3$, so its Hausdorff dimension is $\log(2) / \log(3) \approx 0.631$
• The Sierpiński triangle is the union of three copies of itself, each scaled by a factor of $1/2$, so its Hausdorff dimension is $\log(3) / \log(2) \approx 1.585$
• The Koch curve is the union of four copies of itself, each scaled by a factor of $1/3$, so its Hausdorff dimension is $\log(4) / \log(3) \approx 1.262$
• The Sierpiński carpet is the union of eight copies of itself, each scaled by a factor of $1/3$, so its Hausdorff dimension is $\log(8) / \log(3) \approx 1.893$

Dimension spectrum and applications

Concept of dimension spectrum

• The dimension spectrum is a function that assigns a dimension to each point of a set based on the local scaling properties around that point
• Provides a more refined characterization of the set's irregularity and heterogeneity compared to the global Hausdorff dimension
• Related to the concept of multifractal analysis, which studies the distribution of local scaling exponents in a set or measure

Applications

• Analysis of turbulence: The dimension spectrum can help characterize the intermittency and multi-scale nature of turbulent flows
• Financial time series: The dimension spectrum can reveal the presence of different scaling behaviors in financial data (stock prices, exchange rates) and aid in risk assessment
• Medical images: The dimension spectrum can be used to quantify the complexity and heterogeneity of medical images (MRI, CT scans) and assist in diagnosis and treatment planning
• Geophysical data: The dimension spectrum can help analyze the fractal properties of geophysical data (seismic data, satellite imagery) and improve the understanding of underlying geological processes

Hausdorff dimension vs other dimensions

Topological dimension

• Topological dimension, such as the covering dimension or the inductive dimension, is based on the local properties of a set and takes integer values
• Does not capture the set's scaling properties or irregularities
  • Example: The topological dimension of the Cantor set is 0, while its Hausdorff dimension is $\log(2) / \log(3) \approx 0.631$

Box-counting dimension

• Box-counting dimension, also known as Minkowski dimension or fractal dimension, is based on the asymptotic behavior of the number of boxes needed to cover a set at different scales
• Easier to estimate numerically than Hausdorff dimension but may not always coincide with it
  • Example: For the Cantor set, the box-counting dimension and Hausdorff dimension coincide, but for the Sierpiński carpet, the box-counting dimension is $\log(8) / \log(3) \approx 1.893$, while the Hausdorff dimension is $\log(8) / \log(3) - \log(4) / \log(9) \approx 1.864$

Packing dimension

• Packing dimension is dual to Hausdorff dimension and is defined using the concept of packing measure
• Provides an upper bound for the Hausdorff dimension and coincides with it for many sets
  • Example: For the Cantor set and the Sierpiński triangle, the packing dimension and Hausdorff dimension coincide

Comparison

• Hausdorff dimension is the most refined among these notions, as it captures the set's scaling properties and irregularities, and takes into account the size of covering sets in its definition
• Topological dimension $\leq$ Hausdorff dimension $\leq$ Packing dimension
• Box-counting dimension and Hausdorff dimension may coincide for some sets but not always
  • Example: For the Cantor set, all four dimensions (topological, Hausdorff, box-counting, and packing) coincide, but for the Sierpiński carpet, they are different