Calculating Hausdorff Dimension in Higher Dimensional Spaces

Abstract

In this paper, we prove the identity dimH(F)=d⋅dimH(α−1(F)) , where dimH denotes Hausdorff dimension, F⊆Rd , 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-Rafajłowicz and García-Mora-Redtwitz theorems.