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Iterated Function Systems Enriched with Symmetry

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Abstract

An iterated function system (IFS) can be enriched with an isometry in such a way that the resulting fractal set has prescribed symmetry. If the original system is contractive, then its associated self-similar set is an attractor. On the other hand, the enriched system is no longer contractive and therefore does not have an attractor. However, it posses a self-similar set which, under certain conditions, behaves like an attractor. We give a rigorous procedure which relates a given enriched IFS to a contractive one. Further, we link this procedure to the Lasota–Myjak theory of semiattractors, and so via invariant measures to probabilistic iterated function systems. The chaos game algorithm for enriched IFSs is discussed. We illustrate our main results with several examples which are related to classical fractals such as the Sierpiński triangle and the Barnsley fern.

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Acknowledgements

We would like to thank Filip Strobin for fruitful discussions, and especially for pointing out to us that the results of our paper hold not only for proper spaces but for complete spaces as well. We also thank the unknown referees for their valuable comments which improved the presentation of our paper.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, Chopina 12/18, 87-100, Toruń, Poland

    Krzysztof Leśniak

  2. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

    Nina Snigireva

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  1. Krzysztof Leśniak
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  2. Nina Snigireva
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Correspondence to Nina Snigireva.

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Communicated by Jeffery Geronimo.

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Leśniak, K., Snigireva, N. Iterated Function Systems Enriched with Symmetry. Constr Approx 56, 555–575 (2022). https://doi.org/10.1007/s00365-021-09560-3

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