f(α) spectrum, Rényi dimensions D_q, and multifractal measures
Measure type
Bias p₁0.30
q range5
Resolution6
Multifractals have scale-dependent complexity: different regions scale with different local Hölder exponents α. Generalized (Rényi) dimensions: D_q = lim_{ε→0} (1/(q-1)) log(Σ pᵢ^q) / log(ε). D_0 = box-counting dim, D_1 = information dim, D_2 = correlation dim. f(α) spectrum: Legendre transform of (q-1)D_q gives the fractal dimension f(α) of the set of points with Hölder exponent α.
For a monofractal: f(α) is a single point. For a multifractal: f(α) is a concave curve (multifractal spectrum).