Multifractal system

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A multifractal system is a generalization of a fractal system in which a single exponent, the fractal dimension is not enough to describe its dynamics, and a continuous spectrum of exponents (the so called singularity spectrum) is needed.

In a multifractal system s, the behavior around any point is described by a local power law:

The exponent is called singularity exponent, as it describes the local degree of singularity or regularity around the point .

The ensemble formed by all the points that share the same singularity exponent is called singularity manifold of exponent h, and is a fractal set of fractal dimension D(h). The curve D(h) vs. h is called singularity spectrum and fully describes the (statistical) distribution of the variable s.

Multifractal systems are common in nature, specially Geophysics. They include Fully Developed Turbulence, Stock market time series, real world scenes, Sun’s magnetic field time series, heartbeat dynamics, human gait or natural luminosity time series.