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The Existence of the Attractor of Countable Iterated Function Systems

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Abstract

The theory of iterated function systems (IFS) and of infinite iterated function systems consisting of contraction mappings has been studied in the last decades. Some extensions of the spaces and the contractions concern many authors in fractal theory. In this paper there are described some results in that topic concerning the existence and uniqueness of nonempty compact set which is a set ”fixed point” of a countable iterated function system (CIFS). Moreover, some approximations of the attractor of a CIFS by the attractors of the partial IFSs are given.

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References

  1. Barnsley M.: Fractals everywhere. Academic Press Professional, Boston (1993)

    MATH  Google Scholar 

  2. Berinde V.: Generalized Contractions and Applications. Cub Press 22, Baia-Mare (1997)

    MATH  Google Scholar 

  3. Chiţescu I., Miculescu R.: Approximation of fractals generated by Fredholm integral equations. J. Comput. Anal. Appl. 11(2), 286–293 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Dugundji J., Granas A.: Fixed Point Theory. Springer Verlag, Berlin (1993)

    Google Scholar 

  5. Dumitru D.: Generalized iterated function systems containing Meir-Keeler functions. An. Univ. Bucureşti, Math. LVIII, 3–15 (2009)

    Google Scholar 

  6. Edelstein M.: On fixed and periodic points under contractive mappings. J. London Math. Soc. 37, 74–79 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fernau H.: Infinite Iterated Function Systems. Math. Nachr. 170, 79–91 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gwóźdź-Łukowska G., Jackymski J.: The Hutchinson-Barnsley theory for infinite iterated function systems. Bull. Aust. Math. Soc. 72(3), 441–454 (2005)

    Article  Google Scholar 

  9. Hutchinson J.: Fractals and self-similarity. Indiana Univ. J. Math. 30, 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Käenmäki A.: On natural invariant measures on generalized iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2), 419–458 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Klimek M., Kosek M.: Generalized iterated function systems, multifunctions and Cantor sets. Ann. Pol. Math. 96(1), 25–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Leśniak K.: Infinite Iterated Function Systems: a multivalued approach. Bull. Pol. Acad. Sci. Math. 52(1), 1–8 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Llorens-Fuster E., Petruşel A., Yao J.-C.: Iterated function systems and wellpossedness. Chaos. Solitons and Fractals 41, 1561–1568 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Łoziźski, K. Zyczkowski, W. Słomczyński, Quantum iterated function systems, Phys. Rev. E (3) 68 (2003), No.4, 046110, 9 pp. MR 2060853 (2005a:82065)

  15. Meir A., Keeler E.: A theorem of contractions mappings. J. Math. Anal. Appl. 28, 326–329 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mendivil F.: A generalization of IFS with probabilities to infinitely many maps. J. Rocky Mt. J. Math. 28(3), 1043–1051 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Miculescu R., Mihail A.: Lipscomb’s space ω A is the attractor of an IFS containing affine transformation of l 2(A). Proc. Am. Math. Soc. 136(2), 587–592 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mihail A.: The shift space of a recurrent iterated functions systems. Rev. Roum. Math. Pures Appl. 53(4), 339–355 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Mihail A.: Recurrent iterated functions systems. Rev. Roum. Math. Pures Appl. 53(1), 43–53 (2008)

    MathSciNet  MATH  Google Scholar 

  20. A. Mihail, R. Miculescu, Applications of Fixed Point Theorems in the Theory of Generalized IFS, Fixed Point Theory Appl., 2008, Article ID 312876, 11 pages, doi:10.1155/312876

  21. Mihail A., Miculescu R.: The shift space for an infinite iterated function systems. Math. Rep. 11(1), 21–32 (2009)

    MathSciNet  Google Scholar 

  22. Mihail A., Miculescu R.: A generalization of the Hutchinson measure. Mediterr. J. Math. 6(2), 203–213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Petruşel A.: Dynamical systems, fixed points and fractals. Pure Math. Appl. 13(1-2), 275–281 (2002)

    MathSciNet  Google Scholar 

  24. I.A.Rus, Generalized contractions and Applications, Cluj University Press, 2001

  25. I. Rus, A.Petruşel, G.Petruşel, Fixed Point Theory, Cluj University Press, 2008

  26. Secelean N.A.: Countable Iterated Function Systems. Far East J. Dyn. Syst., Pushpa Publishing House 3(2), 149–167 (2001)

    MathSciNet  MATH  Google Scholar 

  27. N.A. Secelean, Any compact subset of a metric space is the attractor of a CIFS, Bull. Math. Soc. Sc. Math. Roumanie, tome 44 (92), nr.3 (2001), 77-89

  28. N.A. Secelean, Măsurăşi Fractali [Measure and Fractals], ”Lucian Blaga” University Press, Sibiu, 2002

  29. N.A. Secelean, Some continuity and approximation properties of a countable iterated function system, Math. Pannonica, 14/2 (2003), 237-252

    Google Scholar 

  30. Tarski A.: A lattice-theoretical fixed point theorem and its applications. Pacific J. Math. 5, 285–309 (1955)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, “Lucian Blaga” University, str. Dr. Ion Ratiu, nr. 5-7, 550012, Sibiu, Romania

    Nicolae Adrian Secelean

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  1. Nicolae Adrian Secelean
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Secelean, N.A. The Existence of the Attractor of Countable Iterated Function Systems. Mediterr. J. Math. 9, 61–79 (2012). https://doi.org/10.1007/s00009-011-0116-x

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  • DOI: https://doi.org/10.1007/s00009-011-0116-x

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