Calculating Hausdorff Dimension in Higher Dimensional Spaces

Abstract

In this paper, we prove the identity dim H(F) = d dim H(a?1(F)), where dim H denotesHausdorff dimension, F Rd, and a : [0, 1] ! [0, 1]d is a function whose constructive definition isaddressed from the viewpoint of the powerful concept of a fractal structure. Such a result standsparticularly from some other results stated in a more general setting. Thus, Hausdorff dimension ofhigher 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 calculationof the fractal dimension in applications by applying a procedure contributed by the authors inprevious works. It is also worth pointing out that our results generalize both Skubalska-Rafajłowiczand García-Mora-Redtwitz theorems.