📂Dynamics

Hausdorff Dimension

Definition ¹

Assume a metric space $\left( X, d \right)$ is given. The diameter of $S \subset X$ $\diam S$ is defined as follows. $$\diam S := \sup \left\{ d (x, y) : x, y \in S \right\}$$

Hausdorff Outer Measure

Let $S$ be a subset of $X$. For a positive $\delta > 0$, if the union of $U_{k}$ whose diameters are less than $\delta$ $\cup_{k=1}^{\infty} U_{k}$ is a countable covering of $S$, then $H_{\delta}^{d}$ for $d \ge 0$ is defined as follows. $$H_{\delta}^{d} \left( S \right) := \inf \left\{ \sum_{k=1}^{\infty} \left( \diam U_{k} \right)^{d} : \bigcup_{k=1}^{\infty} U_{k} \supset S \land \diam U_{k} < \delta \right\}$$ And for this, the $d$-dimensional Hausdorff outer measure $H_{\delta}^{d}$ is defined as follows. $$H^{d} \left( S \right) := \lim_{\delta \to 0} H_{\delta}^{d} \left( S \right)$$

Hausdorff Dimension

The Hausdorff dimension of $S$ is defined as follows. $$\dim \left( S \right) := \inf \left\{ d \ge 0 : H^{d} (S) = 0 \right\}$$

Explanation

The reason for defining such dimensions using measure theory, which often falls outside the interest of non-mathematics majors, is to ‘measure’ the size of sets with complex structures, such as universally self-similar sets.

The Hausdorff dimension can be seen as the prototype of the box-counting dimension. Although the definition of the Hausdorff dimension itself appears too abstract to intuitively grasp, understanding its significance from a theoretical perspective becomes clear after delving into discussions around fractals and more.

See Also

• Fractals
• Hausdorff Dimension
• Similarity Dimension
• Box-Counting Dimension
• Correlation Dimension