Fractal dimension for fractal structures: A Hausdorff approach revisited

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties.One of these definitions is in terms of finite coverings by elements of the fractal structure. We prove that this dimension is equal to the Hausdorff dimension for compact subsets of Euclidean spaces. This may be the key for the creation of new algorithms to calculate the Hausdorff dimension of these kinds of space.