FRACTAL TOPOGRAPHY AND COMPLEXITY ASSEMBLY IN MULTIFRACTALS

Multifractals are the general form of scale invariances, where a fractal behavior is repeated by a similar object or pattern. Although the multifractal spectrum has been accepted as a measure of behavioral complexity, it cannot solely determine a fractal behavior, thus leaving the control mechanisms of multifractality unclarified. Here, we reexamine the multifractality, and discover two key features of scaling behaviors in the fractal behavior, i.e. multiplicity and repetition. Afterwards, we establish multifractal topography to unify the scale-invariance definition of arbitrary fractal behavior, clarify the physical meaning of singularity and its significance, construct a scale-invariance tree to delineate the control mechanisms in fractality, and then identify multifractals to be dual-complexity systems with the original and behavioral complexities controlling the scaling type and the scale-invariance property independently. This study gives insight into quantitative fractal theory and provides fundamental support for the mechanism exploration of nonlinear dynamics.