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Calculating Hausdorff Dimension in Higher Dimensional Spaces

1
University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720 Santiago de la Ribera, Murcia, Spain
2
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203 Cartagena, Murcia, Spain
3
Departamento de Matemáticas, Universidad de Almería, 04120 La Cañada de San Urbano, Almería, Spain
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(4), 564; https://doi.org/10.3390/sym11040564
Received: 4 April 2019 / Revised: 16 April 2019 / Accepted: 17 April 2019 / Published: 18 April 2019
(This article belongs to the Special Issue New Trends in Dynamics)
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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. View Full-Text
Keywords: Hausdorff dimension; fractal structure; space-filling curve Hausdorff dimension; fractal structure; space-filling curve
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Fernández-Martínez, M.; García Guirao, J.L.; Sánchez-Granero, M.Á. Calculating Hausdorff Dimension in Higher Dimensional Spaces. Symmetry 2019, 11, 564.

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