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Irreducible fractal structures for Moran type theorems

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  • Sánchez-Granero, M.A.
  • Fernández-Martínez, M.

Abstract

In this paper, we introduce a separation property for self-similar sets which is necessary to reach the equality between the similarity dimension and the Hausdorff dimension of these spaces. The similarity boundary of a self-similar set is investigated from the viewpoint of that property. In this way, the strong open set condition (in the self-similar set setting) posed by Keesling and Krishnamurthi has been weakened leading to a Moran type theorem. Moreover, both a result based on a conjecture posed by Deng and Lau as well as an improved version of a theorem due to Bandt and Rao have been contributed regarding the size of the overlaps among the pieces of a self-similar set. Several (equivalent) conditions leading to the equality between the similarity dimension and a new Hausdorff type dimension for attractors described in terms of finite coverings are also provided. Finally, we list some open questions.

Suggested Citation

  • Sánchez-Granero, M.A. & Fernández-Martínez, M., 2019. "Irreducible fractal structures for Moran type theorems," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 29-36.
    • Handle: RePEc:eee:chsofr:v:119:y:2019:i:c:p:29-36
    DOI: 10.1016/j.chaos.2018.12.009
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    References listed on IDEAS

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    1. Bandt, Christoph & Barnsley, Michael & Hegland, Markus & Vince, Andrew, 2016. "Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 478-489.
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