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On Hyperbolic Affine Generalized Infinite Iterated Function Systems

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Abstract

The aim of this paper is to provide alternative characterizations of hyperbolic affine generalized infinite iterated function systems. More precisely, we prove that, for such a system \({\mathcal {F}}=((X,\left\| .\right\| ),(f_{i})_{i\in I})\), among others, the following statements are equivalent: (a) \({\mathcal {F}}\) is hyperbolic. (b) \( {\mathcal {F}}\) has attractor. (c) \({\mathcal {F}}\) is strictly topologically contractive. (d) \({\mathcal {F}}\) is uniformly point-fibred. In this way we generalize the result from the paper by Miculescu and Mihail (J Math Anal Appl 407:56–68, 2013). More equivalent statements are given for the particular case when I is finite and X is finite dimensional.

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Acknowledgements

The authors are very grateful to the reviewer’s valuable remarks and comments which lead to a substantial improvement of the paper.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science University of Bucharest, Romania, Academiei Street 14, 010014, Bucharest, Romania

    Alexandru Mihail

  2. Faculty of Mathematics and Computer Science University of Piteşti, Romania, Târgul din Vale 1, 110040, Piteşti, Argeş, Romania

    Silviu-Aurelian Urziceanu

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  1. Alexandru Mihail
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  2. Silviu-Aurelian Urziceanu
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Correspondence to Silviu-Aurelian Urziceanu.

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Mihail, A., Urziceanu, SA. On Hyperbolic Affine Generalized Infinite Iterated Function Systems. Results Math 75, 111 (2020). https://doi.org/10.1007/s00025-020-01232-1

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