A real attractor non admitting a connected feasible open set

Manuel Fernández-Martínez

doi:10.3934/dcdss.2019046

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2019, Volume 12, Issue 4&5: 723-725. Doi: 10.3934/dcdss.2019046 open access

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A real attractor non admitting a connected feasible open set

Manuel Fernández-Martínez^,

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

* Corresponding author: M. Fernández-Martínez
* Corresponding author: M. Fernández-Martínez

Received: August 31, 2017

Revised: December 31, 2017

Published: March 2019

The author has been partially supported by grants No. MTM2014-51891-P, No. MTM2015- 64373-P (both of them from Spanish Ministry of Economy and Competitiveness), and grant No. 19219/PI/14 from Fundación Séneca of Región de Murcia

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Abstract

A self-similar set is described as the unique (nonempty) compact subset remaining invariant under the action of a finite collection of similitudes on a complete metric space. Among this kind of fractals, those satisfying the so-called Moran's open set condition are especially appropriate to deal with applications of Fractal Geometry since their Hausdorff dimensions can be easily computed. However, such a separation property depends on an external open set whose properties are not fully known. In this paper, we construct a self-similar set in the real line lying under the open set condition which does not admit a connected feasible open set. This answers an open question posed by Zhou and Li in 2009.

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Mathematics Subject Classification: Primary: 28A80.

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References

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