Download Wolfram Notebook

An iterated function system is a nonempty finite set of contraction mappings for , 2, ..., , where , each with a contractivity factor , mapping a nonempty compact metric space into itself. It is the basis for fractal image compression techniques.

The system has a unique nonempty compact set , called its attractor, satisfying

Repeatedly applying all the maps to a nonempty compact set converges to in the Hausdorff metric (Hutchinson 1981).

For the three vertices of an equilateral triangle, the maps have the Sierpiński sieve as their attractor. The illustration shows the three images of the initial triangle and a seventh-stage approximation, colored by the first map. The golden gaskets use the same three centers with specially chosen contraction ratios greater than , producing overlapping images (Broomhead et al. 2004). The two maps that generate the Goffinet dragon combine a common rotation and scaling with different translations in the complex plane.

The chaos game samples the same fixed system by choosing an index independently at each step and setting . For positive probabilities with , the stationary probability measure satisfies

where is a Borel set and denotes its pre-image. Changing these probabilities can change the measure while leaving its support unchanged (Hutchinson 1981). When the maps are similarities, this is a self-similar measure.

See also

Barnsley's Fern, Chaos Game, Contraction Mapping, Exact Overlaps Conjecture, Koch Curve, Self-Similar Measure, Self-Similarity, Sierpiński Sieve

Explore with Wolfram|Alpha

References

Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993.Barnsley, M. F. "Fractal Image Compression." Not. Amer. Math. Soc. 43, 657-662, 1996.Barnsley, M. F. and Demko, S. G. "Iterated Function Systems and the Global Construction of Fractals." Proc. Roy. Soc. London, Ser. A 399, 243-275, 1985.Barnsley, M. F. and Hurd, L. P. Fractal Image Compression. Wellesley, MA: A K Peters, 1993.Bogomolny, A. "The Collage Theorem." https://www.cut-the-knot.org/ctk/ifs.shtml.Broomhead, D.; Montaldi, J.; and Sidorov, N. "Golden Gaskets: Variations on the Sierpiński Sieve." Nonlinearity 17, 1455-1480, 2004. https://doi.org/10.1088/0951-7715/17/4/017.Diaconis, P. M. and Shashahani, M. "Products of Random Matrices and Computer Image Generation." Contemp. Math. 50, 173-182, 1986.Fisher, Y. Fractal Image Compression. New York: Springer-Verlag, 1995.Hutchinson, J. E. "Fractals and Self-Similarity." Indiana Univ. Math. J. 30, 713-747, 1981. https://doi.org/10.1512/iumj.1981.30.30055.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. §1.1.1. https://www.mathematicaguidebooks.org/.Wagon, S. "Iterated Function Systems." §5.2 in Mathematica in Action. New York: W. H. Freeman, pp. 149-156, 1991.

Referenced on Wolfram|Alpha

Iterated Function System

Cite this as:

Weisstein, Eric W. "Iterated Function System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IteratedFunctionSystem.html

Subject classifications