Hausdorff Dimension

p-dimensional Hausdorff Measure

Let $A\subset {\mathbb{R}}^n$ be a bounded subset, fix $p\ge 0$, for $ϵ>0$, let $M(A,p,ϵ)={\inf }_{\text{all covers}}\{{\sum }_{i=1}^{\infty }(\text{diam}(A_i){)}^p\mid \{A_i\}\in A_{ϵ}\text{and}\text{diam}(A_i)<ϵ\},$ where $A_{ϵ}$ =$\{$ countable $ϵ$-cover of $A$ $\}$, $\text{diam}(A_i)={\sup }_{x,y\in A_i}\{d(x,y)\}$. The p-dimensional Hausdorff measure $M(A,p)$ is defined as: $M(A,p)={\lim }_{ϵ\to 0}M(A,p,ϵ).$

Lemma

Let $S:{\mathbb{R}}^n\to {\mathbb{R}}^n$ the a similitude of scale factor $\lambda >0$, then $M(S(A),p)={\lambda }^{p}M(A,p).$

Hausdorff Dimension

The Hausdorff dimension of $A$ is the critical value $s$ that $M(A,p)=\{\begin{array}{ll}\infty & \text{if }0\le p<s\\ c& \text{if }p=s\\ 0& \text{if }p>s\end{array}.$

Proposition

• Monotonicity: If a set $E$ is completely contained within another set $F$, the dimension of $E$ cannot exceed the dimension of $F$. $E\subset F⟹\dim (E)\le \dim (F)$
• Countable Stability: The dimension of a countable union of sets is equal to the supremum (the least upper bound) of the dimensions of the individual sets. $\dim \left({\bigcup }_{i=1}^{\infty }{F}_i\right)=\sup \{\dim (F_i)\}$
• Countable Sets: If a set $F$ consists of only a countable number of points (like the integers or rational numbers), its dimension is exactly zero. $\text{If }F\text{ is a countable set, }\dim (F)=0$

Theorem

If an IFS, $({\mathbb{R}}^m,f_1,\dots f_k)$, is totally disconnect or just-touching, $f_i$ are similitude with $s_i$ the scale factor, then the Hausdorff dimension $H$ of attractor $A$ is the solution of equation: ${\sum }_{i=1}^n|s_i{|}^H=1.$ That is the Hausdorff dimension is equal to the box-counting dimension.