Many fractals are multifractal: different regions have different local scaling dimensions. The multifractal spectrum $f(\alpha)$ describes the distribution of local dimensions. For a measure $\mu$ on a fractal:

• $\alpha$ is the local H\u00f6lder exponent (local scaling)
• $f(\alpha)$ is the Hausdorff dimension of the set where the local dimension equals $\alpha$

The spectrum $f(\alpha)$ versus $\alpha$ is typically concave, with maximum at the global dimension. Multifractal analysis reveals heterogeneity in attractor structure invisible to single-dimension measures.

For a measure $\mu$ supported on a set $F$, cover $F$ with boxes of size $\epsilon$ and let $p_i(\epsilon)$ be the measure of the $i$-th box. The information dimension is:

$d_I = \lim_{\epsilon \to 0} \frac{-\sum_i p_i \log p_i}{\log \epsilon}$

This dimension weights boxes by their measure, capturing information content. For uniform measures, $d_I = \dim_B$, but for non-uniform measures, $d_I \leq \dim_B$ reflects concentration.

Given $N$ points on an attractor, count pairs within distance $r$: $C(r) = \lim_{N \to \infty} \frac{1}{N^2} \#\{(i,j) : |x_i - x_j| < r\}$. The correlation dimension is:

$d_C = \lim_{r \to 0} \frac{\log C(r)}{\log r}$

This dimension is computable from time-series data, making it practical for experimental systems where only scalar measurements are available (via embedding theorems).