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(21 citation statements)

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“…More information is provided about the space when deeper levels in its fractal structure are explored. For instance, note that some kind of common fractal objects like self-similar sets are equipped with a fractal structure in a natural way (see [28], Def. 4.4) as well as it happens with any euclidean space R d which also has a natural fractal structure consisting of d-dimensional cubes of side 1 2 n on the nth-level of its fractal structure (see [29], Def.…”

confidence: 99%

“…More information is provided about the space when deeper levels in its fractal structure are explored. For instance, note that some kind of common fractal objects like self-similar sets are equipped with a fractal structure in a natural way (see [28], Def. 4.4) as well as it happens with any euclidean space R d which also has a natural fractal structure consisting of d-dimensional cubes of side 1 2 n on the nth-level of its fractal structure (see [29], Def.…”

confidence: 99%

“…Definition 2. 4. Let Γ be a fractal structure on a metric space (X, ρ), F be a subset of X, and assume that δ(F, Γ n ) → 0.…”

confidence: 99%

“…Counterexample 6. 4. There exists an Euclidean IFS F, whose associated IFS-attractor K, equipped with its natural fractal structure Γ as a self-similar set, satisfies that…”

confidence: 99%

“…This bas been carried out classically through the box dimension (see, e.g., [19,Subsection 8.4]), mainly due to the easiness when being calculated or estimated in empirical applications, though the Hausdorff dimension, which is the oldest and also the most accurate (by definition) model for fractal dimension has been considered, too, at least from a theoretical point of view. Recently, novel alternatives, with some desirable analytical properties as it happens with the Hausdorff dimension model, and being as easy to calculate as the Email addresses: magdalena.nowak805@gmail.com (Magdalena Nowak), fmm124@gmail.com (M. Fernández-Martínez) 1 The first author was partially supported by National Science Centre grant DEC-2012/07/N/ST1/03551 2 The second author specially acknowledges the valuable support provided by Centro Universitario de la Defensa en la Academia General del Aire de San Javier (Murcia, Spain). box dimension, have been contributed.…”

confidence: 99%

“…Another interesting topic that fractal dimension allows to deal with is about the study of self-similar sets, which become a special kind of fractal sets which could be equipped in a quite natural way by a fractal structure (see [1,Definition 4.4]). To calculate the fractal dimension of this kind of fractals through a explicit formula constitutes a worthmentioning task, and this has been explored previously through both the box-counting and the Hausdorff fractal dimensions, though a certain restrictive hypothesis is required (see [18]).…”

confidence: 99%