Multifractals

Multifractals describe systems whose scaling rules vary across locations. They generalize fractals by allowing a whole spectrum of local scaling exponents rather than a single exponent. They are used to model complex phenomena where intense fluctuations occur at many scales, such as turbulence or financial time series.

Mathematically, one studies a measure μ supported on a set in R^d. For small boxes of side ε, let μ_i be the measure in box i. The local Hölder (or singularity) exponent α at a point reflects how μ scales with ε: μ(B(x,ε)) ~ ε^{α(x)}. The collection of α values that occur has a singularity spectrum f(α), defined as the Hausdorff dimension of the set of points with exponent α. The spectrum is obtained from a family of scaling exponents τ(q) via Z(q,ε) = sum_i μ_i^q ~ ε^{τ(q)}. Then α(q) = dτ/dq and f(α(q)) = q α(q) - τ(q).

Multifractality can arise from heterogeneous measures or processes, often modeled by multiplicative cascades or random cascades. Many multifractal models produce a non-linear τ(q) and a broad f(α) curve. In practice, the spectrum is typically convex and supported on a finite interval.

Estimation approaches include box-counting, multifractal detrended fluctuation analysis (MFDFA), and wavelet-based methods such as WTMM. These techniques infer τ(q) or f(α) from data. Applications span turbulence, geophysics, economics and finance, physiology, climate data, network traffic, and image analysis.