A multifractal is a set or measure whose local scaling behavior requires a spectrum of exponents rather than a single
fractal dimension. For a measure 
, the local exponent 
is defined by
 |
when the limit exists, where 
is the ball of radius 
centered at 
and 
for all sufficiently small 
. This expresses power-law scaling without requiring 
to tend to 1. The multifractal spectrum
is the Hausdorff
dimension of the set of points having exponent 
. A monofractal is the special case in which one exponent
suffices.
