How to calculate the Hausdorff dimension using fractal structures

Fernández–Martínez, M.; Sánchez-Granero, M.A.

doi:10.1016/j.amc.2015.04.059

Cited by 18 publications

(16 citation statements)

References 27 publications

(36 reference statements)

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“…Such a result is interesting in itself since it enables Hausdorff dimension to be used in computational applications involving fractal dimension. To deal with this, the algorithm contributed in ( [12], Section 3.1) becomes the key to estimating Hausdorff dimension in 1-dimensional subsets.…”

Section: Discussionmentioning

confidence: 99%

“…Throughout this paper, we have been focused on calculating the fractal dimension of a subset F ⊆ Y in terms of the fractal dimension of its pre-image Theorems 7,9,12,16,17,19,21,and 23). In this section, we state a powerful result (c.f.…”

Section: Remarkmentioning

confidence: 99%

“…Accordingly, the previous result guarantees that Hausdorff dimension of each compact subset F of [0, 1] d can be calculated in terms of the fractal dimension IV of α −1 (F) ⊆ [0, 1]. Thus, the Algorithm provided in [12] may be applied with this aim.…”

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confidence: 98%

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Calculating Hausdorff Dimension in Higher Dimensional Spaces

2019

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Calculating Hausdorff Dimension in Higher Dimensional Spaces

2019

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“…The last step is to provide three models for a fractal dimension definition (with respect to any fractal structure) following the spirit of the Hausdorff dimension. Thus, while the first one is specially interesting, since its description is made in terms of finite coverings (which allowed the authors to contribute the first known overall algorithm to calculate the Hausdorff dimension of any compact Euclidean subset, see [17]), the remaining definitions become close approaches to classical Hausdorff dimension. Hence, upcoming fractal dimension V becomes a discrete version of the latter, whereas fractal dimension VI is proposed in terms of δ-covers.…”

Section: 3mentioning

confidence: 99%

“…However, though its definition is made in terms of finite coverings, it holds that both fractal dimension IV and Hausdorff dimension are equal for compact Euclidean subspaces (see [16,Theorem 3.13 & Corollary 3.14 (2)]). This fact allowed the authors to provide the first known procedure to calculate the Hausdorff dimension in practical applications (see [17,Algorithm 3.1]). It is worth mentioning that fractal dimension IV becomes also an intermediate model between the classical fractal dimension definitions (for additional details, we refer the reader to [16,Remark 3.15]).…”

Section: Monotonicitymentioning

confidence: 99%

Theoretical properties of fractal dimensions for fractal structures

Fernández–Martínez¹

2015

Discrete &Amp; Continuous Dynamical Systems - S

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Hausdorff dimension, which is the oldest and also the most accurate model for fractal dimension, constitutes the main reference for any fractal dimension definition that could be provided. In fact, its definition is quite general, and is based on a measure, which makes the Hausdorff model pretty desirable from a theoretical point of view. On the other hand, it turns out that fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdorff and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdorff dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work.

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