Hausdorff Dimension Type Models for Fractal Structures

Abstract

In this chapter, we shall study how to generalize the Hausdorff dimension for Euclidean sets throughout three new approaches of fractal dimension for a fractal structure. Thus, while two of such fractal dimensions will consist of appropriate discretizations regarding the classical Hausdorff dimension (the so-called fractal dimensions IV and V), the remaining will constitute a continuous approach from the viewpoint of fractal structures (fractal dimension VI). In this way, several results allowing to connect the new dimensions among them are provided. Also, we shall prove some theoretical results allowing to reach the equality among the new models with fractal dimensions I, II, III, and the classical fractal dimensions. Moreover, we shall explore how the analytic construction regarding fractal dimension VI is based on a measure as it is the case of Hausdorff dimension . The key result in this chapter consists of a generalization regarding the classical Hausdorff dimension in the context of Euclidean subsets endowed with their natural fractal structures in terms of both fractal dimensions V and VI. It is also worth mentioning that fractal dimension IV will equal the Hausdorff dimension of compact Euclidean subsets. Interestingly, as a consequence of the latter result, a novel algorithm to calculate the Hausdorff dimension of such a kind of sets will be developed in forthcoming Sect. 4.8. Finally, we shall contribute a Moran’s type theorem allowing to easily calculate these fractal dimensions for IFS-attractors lying under the OSC.