Other meanings of Iterated function system
Mathematics
In mathematics, an iterated function system (IFS) is a finite set of contraction mappings on a complete metric space, typically the plane, whose repeated application generates a fractal set known as the attractor of the system. The concept was formalized by John Hutchinson in 1981 and popularized by Michael Barnsley, who developed the collage theorem for approximating natural shapes. IFSs provide a rigorous framework for understanding self-similarity and are fundamental to fractal geometry, computer graphics, and image compression.
While most IFSs are affine, nonlinear IFSs using polynomial or Möbius transformations can generate a broader class of fractals. For example, the IFS with condensation introduced by Barnsley includes an additional constant map that produces an attractor containing fixed 'condensation' patterns. The concept played a role in the theorem that a space is a continuous image of the Cantor set if and only if it is a compact metric space.
IFSs extend to random IFSs where the maps are chosen according to a probability distribution, linking them to random walks and measure-valued processes. The invariant measure associated with an IFS with probabilities satisfies the Perron-Frobenius equation for the Markov operator; its moments can be computed explicitly in some cases. Notably, the famous Barnsley fern is generated by only four affine maps, one of which is a rotation that produces the stem.5
The collage theorem provides a practical tool for designing IFSs that approximate given images, forming the basis of fractal image compression.
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