Calculating Hausdorff Dimension in Higher Dimensional Spaces
In this paper, we prove the identity dimH(F)=d⋅dimH(α−1(F)) , where dimH 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 othe...
| Main Authors: | , , |
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
| Published: |
MDPI
2020
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| Subjects: | |
| Online Access: | http://hdl.handle.net/10835/7577 |
| _version_ | 1789406561807892480 |
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| author | Fernández Martínez, Manuel García Guirao, Juan Luis Sánchez Granero, Miguel Ángel |
| author_facet | Fernández Martínez, Manuel García Guirao, Juan Luis Sánchez Granero, Miguel Ángel |
| author_sort | Fernández Martínez, Manuel |
| collection | DSpace |
| description | In this paper, we prove the identity dimH(F)=d⋅dimH(α−1(F)) , where dimH 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 García-Mora-Redtwitz theorems. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-7577 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2020 |
| publisher | MDPI |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-75772023-04-12T19:38:30Z Calculating Hausdorff Dimension in Higher Dimensional Spaces Fernández Martínez, Manuel García Guirao, Juan Luis Sánchez Granero, Miguel Ángel Hausdorff dimension fractal structure space-filling curve In this paper, we prove the identity dimH(F)=d⋅dimH(α−1(F)) , where dimH 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 García-Mora-Redtwitz theorems. 2020-01-17T13:16:14Z 2020-01-17T13:16:14Z 2019-04-18 info:eu-repo/semantics/article 2073-8994 http://hdl.handle.net/10835/7577 en https://www.mdpi.com/2073-8994/11/4/564 Attribution-NonCommercial-NoDerivatives 4.0 Internacional http://creativecommons.org/licenses/by-nc-nd/4.0/ info:eu-repo/semantics/openAccess MDPI |
| spellingShingle | Hausdorff dimension fractal structure space-filling curve Fernández Martínez, Manuel García Guirao, Juan Luis Sánchez Granero, Miguel Ángel Calculating Hausdorff Dimension in Higher Dimensional Spaces |
| title | Calculating Hausdorff Dimension in Higher Dimensional Spaces |
| title_full | Calculating Hausdorff Dimension in Higher Dimensional Spaces |
| title_fullStr | Calculating Hausdorff Dimension in Higher Dimensional Spaces |
| title_full_unstemmed | Calculating Hausdorff Dimension in Higher Dimensional Spaces |
| title_short | Calculating Hausdorff Dimension in Higher Dimensional Spaces |
| title_sort | calculating hausdorff dimension in higher dimensional spaces |
| topic | Hausdorff dimension fractal structure space-filling curve |
| url | http://hdl.handle.net/10835/7577 |
| work_keys_str_mv | AT fernandezmartinezmanuel calculatinghausdorffdimensioninhigherdimensionalspaces AT garciaguiraojuanluis calculatinghausdorffdimensioninhigherdimensionalspaces AT sanchezgraneromiguelangel calculatinghausdorffdimensioninhigherdimensionalspaces |