---

Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 409--424 | DOI:10.5890/DNC.2022.09.005

Jose Mathew, Sunil Mathew

Department of Mathematics, National Institute of Technology Calicut, Calicut-673601, Kerala, India

Download Full Text PDF

 

Abstract

Mathematically, Mandelbrot defined fractals as sets with non integer Hausdorff dimension which exceeds topological dimension. Later Hutchinson developed the theory of Iterated Function System (IFS) to explain self similarity mathematically. IFS theory and its generalisations were studied intensively from Barnsly onwards. Different forms of self similarities and their topological properties were discussed. They were carried out to higher dimensional spaces and corresponding results were established in the literature.

References

1. [1]	Karl, W. (1885), {\"Uber die analytische Darstellbarkeit sogenannter willk{\"u}rlicher Functionen einer reellen Ver{\"a}nderlichen}, Sitzungsberichte der K{\"o}niglich Preu{\ss}ischen Akademie der Wissenschaften zu Berlin, 2, 633-639.

2. [2]	Georg, C. (1883), Ueber unendliche, lineare Punktmannichfaltigkeiten, Mathematische Annalen, Springer, 21(4), 545-591.

3. [3]	Peano, G. (1890), Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36(1), 157-160.

4. [4]	Hubert, D. (1891), Ueber die reellen Z{\"u}ge algebraischer curven, Mathematische Annalen, 38(1, 115-138.

5. [5]	von Koch, H. (1904), On a continuous curve tangent constructible from elementary geometry, Arkiv f or Matematik, Astronomi och Fysik, 681-702.

6. [6]	Hausdorff, F. (1919), Dimension und au$\beta$eres ma$\beta$, Mathematische Annalen, 79(1 -2), 157-179.

7. [7]	Hutchinson, J.E. (1981), Fractals and self similarity, Indiana University Mathematics Journal, 30(5), 713-747.

8. [8]	Moran, P.A. (1946, February), Additive functions of intervals and Hausdorff measure, In Mathematical Proceedings of the Cambridge Philosophical Society, 42(1), 15-23.

9. [9]	Mattila, P. (1982), On the structure of self-similar fractals.

10. [10]	Dekking, F.M. (1982), Recurrent sets, Advances in Mathematics, 44(1), 78-104.

11. [11]	Bedford, T. (1986), Dynamics and dimension for fractal recurrent sets, J. London Math. Soc., 33(2), 89-100.

12. [12]	Hata, M. (1985), On the structure of self-similar sets, Japan Journal of Applied Mathematics, 2(2), 381.

13. [13]	Balu, R., Mathew, S., and Secelean, N.A. (2017), Separation properties of (n, m)-IFS attractors, Communications in Nonlinear Science and Numerical Simulation, 51, 160-168.

14. [14]	Williams, R.F. (1971), Composition of contractions, Bulletin of the Brazilian Mathematical Society, 2(2), 55-59.

15. [15]	Kigami, J. (2001), Analysis on fractals, Cambridge University Press, 143.

16. [16]	Bandt, C. and Graf, S. (1992), Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure, Proceedings of the American Mathematical Society, 995-1001.

17. [17]	Schief, A. (1994), Separation properties for self-similar sets, Proceedings of the American Mathematical Society, 122(1), 111-115.

18. [18]	Schief, A. (1996), Self-similar sets in complete metric spaces, Proceedings of the American Mathematical Society, 124(2), 481-490.

19. [19]	Bandt, C. (1991), Self-similar sets 5. Integer matrices and fractal tilings of $R^n$, Proceedings of the American Mathematical Society, 549-562.

20. [20]	Falconer, K.J. (1995), Sub-self-similar sets, Transactions of the American Mathematical Society, 347(8), 3121-3129.

21. [21]	McClure, M. and Vallin, R.W. (2000), The Borel structure of the collections of sub-self-similar sets and super-self-similar sets, Acta Mathematica Universitatis Comenianae. New Series, 69(2), 145-149.

22. [22]	Aswathy, R.K. and Mathew, S. (2016), On different forms of self similarity, Chaos, Solitons and Fractals, 87, 102-108.

23. [23]	Fernau, H. (1994), Infinite iterated function systems, Mathematische Nachrichten, 170(1), 79-91.

24. [24]	Mihail, A. and Miculescu, R. (2009), The shift space for an infinite iterated function system, Math. Rep.(Bucur.), 11(61), 1.

25. [25]	Secelean, N.A. (2001), Any compact subset of a metric space is the attractor of a countable function system, Bulletin math{\"ematique de la Soci{\"e}t{\"e} des Sciences Math{\"e}matiques de Roumanie}, 237-241.

26. [26]	Secelean, N.A. (2012), The existence of the attractor of countable iterated function systems, Mediterranean journal of mathematics, 9(1), 61-79.

27. [27]	Secelean, N.A. (2011), Generalized countable iterated function systems, Filomat, 25(1), 21-36.

28. [28]	Mihail, A. and Miculescu, R. (2008), Applications of fixed point theorems in the theory of generalized IFS, Fixed Point Theory and Applications, 2008(1), 1-11.

29. [29]	Luo, J., Rao, H., and Tan, B. (2002), Topological structure of self-similar sets, Fractals, 10(02), 223-227.

30. [30]	Duvall, P.F. and Husch, L.S. (1992), Attractors of iterated function systems, Proceedings of the American Mathematical Society, 116(1), 279-284.

31. [31]	Balu, R. and Mathew, S. (2013), On (n, m)-iterated function system, Asian-European Journal of Mathematics, 6(04), 1350055.

32. [32]	Aswathy, R.K. and Mathew, S. (2019), Separation properties of finite products of hyperbolic iterated function systems, Communications in Nonlinear Science and Numerical Simulation, 67, 594-599.