q-Dimension

$$D_q\equiv {1\over 1-q} \lim_{\epsilon\to 0} {\ln I(q,\epsilon)\over \ln\left({1\over \epsilon}\right),}$$ (1)

where

$$I(q,\epsilon)\equiv \sum_{i=1}^N {\mu_i}^q,$$ (2)

$\epsilon$ is the box size, and ${\mu_i}$ is the Natural Measure. If $q_1>q_2$, then

$$D_{q_1} \leq D_{q_2}.$$ (3)

The Capacity Dimension (a.k.a. Box Counting Dimension) is given by $q=0$,

$$D_0 = {1\over 1-0} \lim_{\epsilon\to 0} {\ln\left({\sum_{i=1... ...= - \lim_{\epsilon\to 0} {\ln[N(\epsilon)]\over \ln \epsilon}.$$ (4)

If all ${\mu_i}$s are equal, then the Capacity Dimension is obtained for any $q$. The Information Dimension is defined by

$$D_1 = \lim_{q\to 1} D_q = \lim_{q\to 1}{\lim_{\epsilon\to 0}{\ln\left[{\sum_{i=1}^{N(\epsilon)}{\mu_i}^q}\right]\over -\ln\epsilon}\over {1-q}}$$

$$= \lim_{\epsilon\to 0} \lim_{q\to 1} {\ln\left({\sum_{i=1}^{N(\epsilon)} {\mu_i}^q}\right)\over\ln\epsilon(q-1)}.$$ (5)

But

$$\lim_{q\to 1} \ln\left({\sum_{i=1}^{N(\epsilon)} {\mu_i}^q}\... ...= \ln\left({\sum_{i=1}^{N(\epsilon)} \mu_i}\right)= \ln 1 = 0,$$ (6)

so use L'Hospital's Rule

$$D_1 = \lim_{\epsilon\to 0}\left({{1\over\ln\epsilon} \lim_{q\to 1} {\sum q{\mu_i}^{q-1}\over\sum {\mu_i}^q}}\right).$$ (7)

Therefore,

$$D_1 = \lim_{\epsilon\to 0} {\sum_{i=1}^{N(\epsilon)} \mu_i\ln \mu_i\over \ln\epsilon}.$$ (8)

$D_2$ is called the Correlation Dimension. The $q$-dimensions satisfy

$$D_{q+1}\leq D_q.$$ (9)

See also Fractal Dimension