multifractal

Multifractal describes a property of certain measures and signals whose local scaling behavior cannot be captured by a single exponent. In contrast to monofractals, where a single Hölder or fractal dimension suffices, a multifractal object exhibits a range of scaling rules across its domain.

The mathematical framework associates a local Hölder exponent α(x) with almost every point x, defined by the behavior of a measure μ on small balls B_r(x): μ(B_r(x)) ~ r^{α(x)} as r → 0. Collecting points with the same α leads to a singularity spectrum f(α), which gives the Hausdorff dimension of the set {x: α(x)=α}. The spectrum is linked to a mass exponent function τ(q) through the partition function Z(q,r) = Σ_i μ(B_i)^q ~ r^{τ(q)}; the Legendre transform of τ(q) yields f(α) as a function of α. Nonlinearity of τ(q) signals multifractality, whereas linear τ(q) corresponds to a monofractal (single exponent) structure.

Typical methods to estimate the multifractal spectrum from data include the partition function (structure-function) method, wavelet transform modulus maxima (WTMM), and variations of multifractal detrended fluctuation analysis. Finite-size effects and statistical fluctuations are common challenges.

Multifractal models include multiplicative cascades (for example, binomial measures) and lognormal or log-Poisson models, which generate broad spectra for α. The multifractal description has been applied to turbulence, financial time series, internet traffic, rainfall, geophysics, and physiology (such as heart-rate variability).

Historically, multifractal concepts were developed in the late 20th century, with foundational work by Halsey, Jensen, Kadanoff, and Procaccia among others.