Box dimension type models

Abstract

The main goal of this chapter is to generalize the classical box dimension in the broader context of fractal structures. We state that whether the so-called natural fractal structure (which any Euclidean subset can be always endowed with) is selected, then the box dimension remains as a particular case of the generalized fractal dimension models. That idea allows to consider a wider range of fractal structures to calculate the fractal dimension of a given subset. Interestingly, unlike how it happens with classical box dimension, the new models we provide in this chapter can be further extended to non-Euclidean contexts, where the classical definitions of fractal dimension may lack sense or cannot be calculated. In this chapter, we illustrate this fact in the context of the domain of words. Another advantage of these models of fractal dimension for fractal structures lies in the possibility of their effective calculation or estimation for any space admitting a fractal structure. To calculate these dimensions, we can proceed as easy as to estimate the box dimension in Euclidean applications.