Calculating Hausdorff dimension in higher dimensional spaces

Abstract

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-Rafajłowicz and García-Mora-Redtwitz theorems.