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Calculating Hausdorff Dimension in Higher Dimensional Spaces

University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720 Santiago de la Ribera, Murcia, Spain
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203 Cartagena, Murcia, Spain
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;
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)
PDF [537 KB, uploaded 24 April 2019]


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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