Calculating Hausdorff dimension in higher dimensional spaces

Abstract

In this paper, we prove the identity ${\dim }_{\text{H}}(F)=d\cdot {\dim }_{\text{H}}({\alpha }^{-1}(F))$, where ${\dim }_{\text{H}}$ denotes Hausdorff dimension, $F\subseteq {\mathbb{R}}^d$, and $\alpha :[0,1]\to [0,1{]}^d$ 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 Garc\'ıa-Mora-Redtwitz theorems.

20 pages, 3 figures