Box-Counting Dimension

Box-Counting Dimension

Let $E\subset {\mathbb{R}}^n$ a non-empty compact set, $N_E(r)=\text{the minimal number of closed balls }B(x,r)\text{needed to cover}E.$ Then the box-counting dimension $D$ of $E$ is $D={\lim }_{r\to \infty }\frac{\log N_E(r)}{\log \frac{1}{r}}.$

eg. $E=\{\frac{1}{n}\mid n>0\}\cup \{0\}$, then $D=\frac{1}{2}$.

Definition

Let $(X,d)$ be a complete metric space, typically ${\mathbb{R}}^n$ with the standard Euclidean metric. A mapping $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a similitude (or a similarity contraction) if there exists a constant scaling factor $r$ where $0<r<1$, such that for all $x,y\in {\mathbb{R}}^n$: $d(f(x),f(y))=r\cdot d(x,y).$ In Euclidean space, every similitude can be explicitly written in the form: $f(x)=rAx+t,$ where $r$ is the scalar contraction factor ($0<r<1$), $A$ is an orthogonal matrix, representing a rotation or a reflection, $t$ is a vector in ${\mathbb{R}}^n$, representing a translation.

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 box-counting dimension $D$ of attractor $A$ is the solution of equation: ${\sum }_{i=1}^n|s_i{|}^D=1.$ If the IFS is overlapping, then the box-counting dimension of $A$ is bounded above by this $D$.