Abstract
We show that the space called the shark teeth is a topological IFS-attractor, that is, for every open cover of \(X = \bigcup\nolimits_{i = 1}^n {f_i (X)}\), its image under every suitable large composition from the family of continuous functions {f 1, ..., f n } lies in some set from the cover. In particular, there exists a space that is not homeomorphic to any IFS-attractor but is a topological IFS-attractor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hutchinson J., “Fractals and self similarity,” Indiana Univ. Math. J., 30, No. 5, 713–747 (1981).
Barnsley M., Fractals Everywhere, Acad. Press, Boston (1988).
Williams R. F., “Composition of contractions,” Bol. Soc. Brasil. Mat., 2, 55–59 (1971).
Hata M., “On the structure of self-similar sets,” Japan J. Appl. Math., 2, No. 2, 381–414 (1985).
Banakh T. and Nowak M., “A 1-dimensional Peano continuum which is not an IFS attractor,” Proc. Amer. Math. Soc., 141, 931–935 (2013).
Banakh T. and Tuncali M., “Controlled Hahn-Mazurkiewicz theorem and some new dimension functions of Peano continua,” Topology Appl., 154, No. 7, 1286–1297 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2014 Nowak M. and Szarek T.
The authors were supported in part by the National Science Centre of Poland (Grant DEC-2012/07/B/ ST1/03320 (the second author) and Grant DEC-2012/07/N/ST1/03551 (the first author)).
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 2, pp. 364–369, March–April, 2014.
Rights and permissions
About this article
Cite this article
Nowak, M., Szarek, T. The shark teeth is a topological IFS-attractor. Sib Math J 55, 296–300 (2014). https://doi.org/10.1134/S0037446614020128
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446614020128