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Fractal Dimension for IFS-Attractors Revisited

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Abstract

One of the milestones in Fractal Geometry is the so-called Moran’s Theorem, which allows the calculation of the similarity dimension of any strict self-similar set under the open set condition. In this paper, we contribute a generalized version of the Moran’s theorem, which does not require the \(\mathrm{OSC}\) to be satisfied by the similitudes that give rise to the corresponding attractor. To deal with, two generalized versions for the classical fractal dimensions, namely, the box and the Hausdorff dimensions, are explored in terms of fractal structures, a kind of uniform spaces.

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Acknowledgements

The authors acknowledge the partial support of Grants No. 19219/PI/14 from Fundación Séneca of Región de Murcia and No. MTM2014-51891-P from Spanish Ministry of Economy and Competitiveness.

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Authors and Affiliations

  1. University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720, Santiago de la Ribera, Región de Murcia, Spain

    M. Fernández-Martínez & Juan Antonio Vera López

  2. Departamento de Matemática Aplicada y Estadística, Hospital de Marina, Universidad Politécnica de Cartagena, 30203, Cartagena, Región de Murcia, Spain

    J. L. G. Guirao

Authors
  1. M. Fernández-Martínez
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  2. J. L. G. Guirao
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  3. Juan Antonio Vera López
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Correspondence to M. Fernández-Martínez.

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Fernández-Martínez, M., Guirao, J.L.G. & Vera López, J.A. Fractal Dimension for IFS-Attractors Revisited. Qual. Theory Dyn. Syst. 17, 709–722 (2018). https://doi.org/10.1007/s12346-018-0272-5

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