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On the attractor of one-dimensional infinite iterated function systems

Published Online:March 22, 2013pp 87-99https://doi.org/10.1504/IJANS.2013.052767

Abstract

We study the attractor of iterated function systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.

Keywords

iterated function systems, IFS, attractors, second generation IFS

References

  • 1. Barnsley, M.F. (1988). Fractals Everywhere. New York, NY:Academic Press Google Scholar
  • 2. Barnsley, M.F. , Demko, S.G. (1985). ‘Iterated function systems and the global construction of fractals’. Proc. R. Soc. London A. 399, 243-275 Google Scholar
  • 3. Cabrelli, C. , Hare, K. , Molter, U. (2002). ‘Sums of Cantor sets yielding an interval’. J. Aust. Math. Soc.. 73, 3, 405-418 Google Scholar
  • 4. Diaconis, P. , Shahshahani, M. (1986). ‘Products of random matrices and computer image generation’. Contemporary Math.. 50, 173-182 Google Scholar
  • 5. Duvall, P.F. , Husch, L.S. (1992). ‘Attractors of iterated function systems’. Proc. Amer. Math. Soc.. 116, 279-284 Google Scholar
  • 6. Elton, J.H. , Yan, Z. (1989). ‘Approximation of measures by Markov processes and homogeneous affine iterated function systems’. Constr. Appr.. 5, 1, 69-87 Google Scholar
  • 7. Fernau, H. (1994). ‘Infinite iterated function systems’. Math. Nach.. 170, 1, 79-91 Google Scholar
  • 8. Handy, C.R. , Mantica, G. (1990). ‘Inverse problems in fractal construction: moment method solution’. Physica D. 43, 1, 17-36 Google Scholar
  • 9. Hata, M. (1985). ‘On the structure of self-similar sets’. Japan J. Appl. Math.. 2, 2, 381-414 Google Scholar
  • 10. Hutchinson, J. (1981). ‘Fractals and self-similarity’. Indiana J. Math.. 30, 5, 713-747 Google Scholar
  • 11. Kwiecínski, M. (1999). ‘A locally connected continuum which is not an IFS attractor’. Bull. Polish Acad. Sci.. 47, 2, Google Scholar
  • 12. Mantica, G. (1996). ‘A Stieltjes technique for computing Jacobi matrices associated with singular measures’. Constr. Appr.. 12, 4, 509-530 Google Scholar
  • 13. Mantica, G. (2007). ‘Polynomial sampling and fractal measures: I.F.S.-Gaussian integration’. Num. Alg.. 45, 1, 269-281 Google Scholar
  • 14. Mantica, G. (2010). ‘Dynamical systems and numerical analysis: the study of measures generated by uncountable I.F.S.’. Num. Alg.. 55, 2, 321-335 Google Scholar
  • 15. Mantica, G. (2011). ‘Direct and inverse computation of Jacobi matrices of infinite IFS’. arXiv:1102.5219v1 [math.CA] Google Scholar
  • 16. Mantica, G. , Sloan, A. (1989). ‘Chaotic optimization and the construction of fractals’. Complex Systems. 3, 1, 37-72 Google Scholar
  • 17. Mauldin, D. , Urbansky, M. (1996). ‘Dimensions and measures in infinite iterated function systems’. Proc. London Math. Soc.. 73, 3, 105-154 Google Scholar
  • 18. Mendivil, F. (1998). ‘A generalization of IFS with probabilities to infinitely many maps’. Rocky Mountain J. Math.. 28, 3, 1043-1051 Google Scholar
  • 19. Moran, M. (1996). ‘Hausdorff measure of infinitely generated self-similar sets’. Mh. Math.. 122, 4, 387-399 Google Scholar
  • 20. Moran, P.A.P. (1946). ‘Additive functions of intervals and Hausdorff measure’. Proc. Camb. Phil. Soc.. 42, 15-23 Google Scholar
  • 21. Peres, Y. , Schlag, W. , Solomyak, B. (2000). ‘Sixty years of Bernoulli convolutions’. in Fractal Geometry and Stochastics II, Progress in Probability. 46, Basel:Birkhauser , 39-68 Google Scholar
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