Calculating Hausdorff Dimension in Higher Dimensional Spaces

In this paper, we prove the identity dim H ( F ) = d · dim H ( α − 1 ( F ) ) , where dim H denotes Hausdorff dimension, F ⊆ R d , and α : [ 0 , 1 ] → [ 0 , 1 ] d is a function whose constructive definition is addressed from the viewpoint of the powerful concept of a fractal structure. Such a result stands particularly from some other results stated in a more general setting. Thus, Hausdorff dimension of higher dimensional subsets can be calculated from Hausdorff dimension of 1-dimensional subsets of [ 0 , 1 ] . As a consequence, Hausdorff dimension becomes available to deal with the effective calculation of the fractal dimension in applications by applying a procedure contributed by the authors in previous works. It is also worth pointing out that our results generalize both Skubalska-Rafajlowicz and Garcia-Mora-Redtwitz theorems.