Journal
CHAOS SOLITONS & FRACTALS
Volume 119, Issue -, Pages 29-36Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2018.12.009
Keywords
Open set condition; Iterated function system; Weak separation property; Moran's theorem; Self-similar set; Hausdorff dimension
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Funding
- MINECO/FEDER, UE [MTM2015-64373-P]
- Fundacion Seneca of Region de Murcia [19219/PI/14]
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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 Hausdorffdimension 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 Hausdorfftype dimension for attractors described in terms of finite coverings are also provided. Finally, we list some open questions. (c) 2018 Elsevier Ltd. All rights reserved.
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