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A survey on fractal dimension for fractal structures. (English) Zbl 1377.28005

Summary: Along the years, the foundations of fractal geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from asymmetric topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof. M. A. Sánchez-Granero in the context of fractal structures.

MSC:

28A78 Hausdorff and packing measures
28A80 Fractals
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
54E35 Metric spaces, metrizability
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[45] M. Fernández-Martínez and Miguel Ángel López Guerrero, (2015), Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension, Discrete and Continuous Dynamical Systems - Series S, 8, No 6, 1129-1137.; Fernández-Martínez, M.; Miguel Ángel López Guerrero, Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension, Discrete and Continuous Dynamical Systems - Series S, 8, 6, 1129-1137 (2015) · Zbl 1334.28018 · doi:10.3934/dcdss.2015.8.1129
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