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, isIt 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 $-[q\log p+(1-q)\log (1-p)]/\log 2$. When $p\ne 1/2$, different digit frequencies produce different exponents. This is the mechanism by which uneven concentration of mass leads to several scaling laws.