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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 12, Issue 4

Issues

Volume 13 (2015)

Countable contraction mappings in metric spaces: invariant sets and measure

María Barrozo
  • Departamento de Matemática, Facultad de Ciencias Físico-Matemáticas y Naturales, Universidad Nacional de San Luis, Ejército de Los Andes 950, 5700, San Luis, Argentina
  • IMASL-CONICET, Italia 1556, 5700, San Luis, Argentina
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ursula Molter
  • Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428, Capital Federal, Argentina
  • IMAS-CONICET, Ciudad Universitaria, Pabellón 1, 1428, Buenos Aires, Argentina
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Published Online: 2014-01-17 | DOI: https://doi.org/10.2478/s11533-013-0371-0

Abstract

We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1.

Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.

MSC: 28A80; 37C25; 37C70

Keywords: Contraction maps; Countable iterated function system; Invariant set; Invariant measure

  • [1] Bandt C., Self-similar sets. I. Topological Markov chains and mixed self-similar sets, Math. Nachr., 1989, 142, 107–123 http://dx.doi.org/10.1002/mana.19891420107CrossrefGoogle Scholar

  • [2] Barnsley M.F., Demko S., Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 1985, 399(1817), 243–275 http://dx.doi.org/10.1098/rspa.1985.0057CrossrefGoogle Scholar

  • [3] Falconer K.J., The Geometry of Fractal Sets, Cambridge Tracts in Math., 85, Cambridge University Press, Cambridge, 1986 Google Scholar

  • [4] Falconer K., Fractal Geometry, John Wiley & Sons, Chichester, 1990 Google Scholar

  • [5] Hille M.R., Remarks on limit sets of infinite iterated function systems, Monatsh. Math., 2012, 168(2), 215–237 http://dx.doi.org/10.1007/s00605-011-0357-6CrossrefWeb of ScienceGoogle Scholar

  • [6] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747 http://dx.doi.org/10.1512/iumj.1981.30.30055CrossrefGoogle Scholar

  • [7] Kravchenko A.S., Completeness of the space of separable measures in the Kantorovich-Rubinshtein metric, Siberian Math. J., 2006, 47(1), 68–76 http://dx.doi.org/10.1007/s11202-006-0009-6CrossrefGoogle Scholar

  • [8] Mattila P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math., 44, Cambridge University Press, Cambridge, 1995 http://dx.doi.org/10.1017/CBO9780511623813CrossrefGoogle Scholar

  • [9] Mauldin R.D., Infinite iterated function systems: theory and applications, In: Fractal Geometry and Stochastics, Progr. Probab., 37, Birkhäuser, Basel, 1995, 91–110 http://dx.doi.org/10.1007/978-3-0348-7755-8_5CrossrefGoogle Scholar

  • [10] Mauldin R.D., Urbanski M., Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 1996, 73(1), 105–154 http://dx.doi.org/10.1112/plms/s3-73.1.105CrossrefGoogle Scholar

  • [11] Mauldin R.D., Williams S.C., Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc., 1986, 295(1), 325–346 http://dx.doi.org/10.1090/S0002-9947-1986-0831202-5CrossrefGoogle Scholar

  • [12] Mihail A., Miculescu R., The shift space for an infinite iterated function system, Math. Rep. (Bucur.), 2009, 11(61)(1), 21–32 Google Scholar

  • [13] Secelean N.A., The existence of the attractor of countable iterated function systems, Mediterr. J. Math., 2012, 9(1), 61–79 http://dx.doi.org/10.1007/s00009-011-0116-xWeb of ScienceCrossrefGoogle Scholar

About the article

Published Online: 2014-01-17

Published in Print: 2014-04-01


Citation Information: Open Mathematics, Volume 12, Issue 4, Pages 593–602, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0371-0.

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© 2014 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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