International Journal of Mathematics Trends and Technology (IJMTT)
© 2021 by IJMTT Journal
Volume-67 Issue-12
Year of Publication : 2021
Authors : T. Abirami, M. Suresh Kumar


MLA Style: T. Abirami, M. Suresh Kumar. "A STUDY ON ITERATED FUNCTION SYSTEMS" International Journal of Mathematics Trends and Technology 67.12 (2021):47-56. 

APA Style: T. Abirami, M. Suresh Kumar(2021). A STUDY ON ITERATED FUNCTION SYSTEMS International Journal of Mathematics Trends and Technology, 67(12), 47-56.

In this paper, every metric d on a nonempty set X induces a metric h, called Hausdorff metric, on the set K(X) of the collection of all non-empty compact subsets of X. It is a well known fact that the induced metric preserves the completeness and compactness. In this paper, we discuss the existence of attractors, which is generally known as fractals, of iterated function systems using the Hausdorff metric. We also discuss how to construct the attractors of iterated function systems using the fixed points of contraction maps involved in the iterated function systems.


[1] M. F. Barnsley, Fractals everywhere. Academic Press, Harcourt Brace Janovitch, 1988
[2] K. J. Falconer, Fractal geometry: Mathematical foundations and applications. Second edition, John Wiley and Sons, Ltd, 2005.
[3] J. Hutchinson, Fractals and self-similarity. Indiana Univ. J. Math. 30, 1981 (p.713-747)
[4] A. Jadczyk, Quantum Fractals, From Heisenbergs Uncertainty to Barnsleys Fractality. World Scientific, 2014.
[5] K. Lesniak, Infinite iterated function systems: a multivalued approach. Bulletin of the Polish Academy of Sciences Mathematics 52, nr.1, 2004 (p.1-8)
[6] B. B. Mandelbrot, The fractal geometry of nature(Vol. 173). New York: WH freeman, 1983.
[7] A. Mihail, R. Miculescu, Applications of Fixed Point Theorems in the Theory of Generalized IFS. Fixed Point Theory and Applications 2008, Article ID 312876, 11 pages.
[8] N. A. Secelean, Countable Iterated Function System. Far East Journal of Dynamical Systems, Pushpa Publishing House, vol. 3(2), 2001 (p.149-167)
[9] 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 (p.77-89)
[10] N. A. Secelean, Generalized countable iterated function systems. Filomat, 2011, 25(1), 21-36.
[11] N. A. Secelean, Some continuity and approximation properties of a countable iterated function system. Mathematica Pannonica, 14/2, 2003 (p.237-252)
[12] N. A. Secelean, The existence of the attractor of countable iterated function systems. Mediter-ranean journal of mathematics, (2012), 9(1), 61-79.
[13] N. A. Secelean, Generalized iterated function systems on the space l1(X). J. Math. Anal. Appl.410, no. 2 (2014), 847-858.
[14] Yuval Fisher, Fractal Image Compression: Theory and Application. Springer Verlag, New York, 1995

Keywords : Hausdorff metric, Compact, Complete,Contraction, Iterate Function Systems,Attractor.