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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 12, 2012

A class of continua that are not attractors of any IFS

  • Marcin Kulczycki EMAIL logo and Magdalena Nowak
From the journal Open Mathematics

Abstract

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

[1] Barnsley M., Fractals Everywhere, Academic Press, Boston, 1988 Search in Google Scholar

[2] Edelstein M., On fixed and periodic points under contractive mappings, J. Lond. Math. Soc., 1962, 37, 74–79 http://dx.doi.org/10.1112/jlms/s1-37.1.7410.1112/jlms/s1-37.1.74Search in Google Scholar

[3] Hata M., On the structure of self-similar sets, Japan J. Appl. Math., 1985, 2(2), 381–414 http://dx.doi.org/10.1007/BF0316708310.1007/BF03167083Search in Google Scholar

[4] Hutchinson J.E., Fractals and self similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747 http://dx.doi.org/10.1512/iumj.1981.30.3005510.1512/iumj.1981.30.30055Search in Google Scholar

[5] Kwiecinski M., A locally connected continuum which is not an IFS attractor, Bull. Pol. Acad. Sci. Math., 1999, 47(2), 127–132 Search in Google Scholar

[6] Sanders M.J., Non-attractors of iterated fuction systems, Texas Project NexT Journal, 2003, 1, 1–9 Search in Google Scholar

[7] Sanders M.J., An n-cell in ℝn+1 that is not the attractor of any IFS on ℝn+1, Missouri J. Math. Sci., 2009, 21(1), 13–20 10.35834/mjms/1316032676Search in Google Scholar

Published Online: 2012-10-12
Published in Print: 2012-12-1

© 2012 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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