Fractal dimension for fractal structures

摘要

The main goal of this paper is to provide a generalized definition of fractal dimension for any space equipped with a fractal structure. This novel theory generalizes the classical box-counting dimension theory on the more general context of GF-spaces. In this way, if we select the so-called natural fractal structure on any Euclidean space, then the box-counting dimension becomes just a particular case. This idea allows to consider a wide range of fractal structures to calculate the effective fractal dimension for any subset of this space. Unlike it happens with the classical theory of fractal dimension, the new definitions we provide may be calculated in contexts where the box-counting one can have no sense or cannot be calculated. Nevertheless, the new models can be computed for any space admitting a fractal structure, just as easy as the box-counting dimension in empirical applications.