\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A real attractor non admitting a connected feasible open set

  • * Corresponding author: M. Fernández-Martínez

    * Corresponding author: M. Fernández-Martínez

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 28A80.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   G. Cantor , Ueber unendliche, lineare Punktmannichfaltigkeiten, Mathematische Annalen, 21 (1883) , 545-591.  doi: 10.1007/BF01446819.
      F. Hausdorff , Dimension und äußeres Maß, Mathematische Annalen, 79 (1918) , 157-179.  doi: 10.1007/BF01457179.
      D. Hilbert , Über die stetige Abbildung einer Line auf ein Flächenstück, Mathematische Annalen, 38 (1891) , 459-460.  doi: 10.1007/BF01199431.
      J. E. Hutchinson , Fractals and self-similarity, Indiana University Mathematics Journal, 30 (1981) , 713-747.  doi: 10.1512/iumj.1981.30.30055.
      P. A. P. Moran , Additive functions of intervals and Hausdorff measure, Mathematical Proceedings of the Cambridge Philosophical Society, 42 (1946) , 15-23.  doi: 10.1017/S0305004100022684.
      G. Peano , Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36 (1890) , 157-160.  doi: 10.1007/BF01199438.
      A. Schief , Separation properties for self-similar sets, Proceedings of the American Mathematical Society, 122 (1994) , 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1.
      Z. Zhou  and  F. Li , Some problems on fractal geometry and topological dynamical systems, Analysis in Theory and Applications, 25 (2009) , 5-15.  doi: 10.1007/s10496-009-0005-3.
  • 加载中
Open Access Under a Creative Commons license
SHARE

Article Metrics

HTML views(901) PDF downloads(245) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return