2019/03/28 by Sánchez-Granero, M. A., Fernández-Martínez, M.
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1903.11926
In this paper, we prove the identity dim\textrm H(F)=d⋅ dim\textrm H(α-1(F)), where dim\textrm H 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 Garc'ıa-Mora-Redtwitz theorems.