Iterated Function Systems

Iterated function systems (IFS) are a method for constructing fractal sets by repeatedly applying a finite collection of contractive transformations to an initial set. The key theorem, due to Hutchinson, states that such a system has a unique nonempty compact fixed point — the "attractor" of the IFS — which is typically a fractal. This framework unifies many classical fractals: the Sierpinski triangle is the attractor of three contractions, the Cantor set of two.

IFS methods extend beyond pure mathematics into image compression, where the inverse problem — finding the transformations that generate a given image — yields remarkable compression ratios for natural textures. The connection to dynamical systems runs deeper: the attractor of an IFS can be understood as the invariant set of a discrete dynamical system, and its dimension can be computed via pressure formulas from ergodic theory.