= Multifractal measure
A <multifractal measure> has different local scaling behavior at different points. For a <measure> $\mu$ on a <metric space>, a basic local exponent, when the limit exists, is
$$
\alpha(x)=\lim_{r\downarrow0}\frac{\log\mu(B(x,r))}{\log r}.
$$
It describes how rapidly mass near $x$ shrinks with radius. The <Hausdorff dimension> of each level set $\{x:\alpha(x)=a\}$ gives a spectrum rather than a single dimension. The word does not require the <measure> to be invariant, although many examples come from <invariant measures> of <dynamical systems>.
For a concrete construction, repeatedly split $[0,1]$ into halves, giving relative mass $p$ to the left half and $1-p$ to the right, with $0<p<1$. A level-$n$ interval with $k$ left choices has length $2^{-n}$ and mass $p^k(1-p)^{n-k}$. Along a nested sequence with $k/n\to q$, its cylinder scaling exponent is $-[q\log p+(1-q)\log(1-p)]/\log2$. When $p\ne1/2$, different digit frequencies produce different exponents. This is the mechanism by which uneven concentration of mass leads to several scaling laws.
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