One of the simplest discrete dynamical systems are the iterated function systems (IFS) associated to a fixed map f.

Consider a family of contractive maps {F_α} for α in Γ, where Γ is some index set. Define the contractive map Φ : K(X) → K(X) such that Φ(A)={x : there exists i in Γ such that x in F_i(A)}, where K(X) is the set of all nonempty compact sets of X. It follows from the general theory of complete metric spaces that Φ has a fixed point, A, called the attractor or the invariant set associated to Φ.

The fractal images associated to these systems are generated by the attractors of some family of IFS. They can be generated by two ways: a deterministic way or a probabilistic way. The deterministic algorithm consists by starting with an arbitrary point and apply all the mappings associated to the family. Each mapping returns a point that is drawn. Each point is then applied to each mapping again, returning more points. But eventually there will be too many points to store in memory and, thus, this algorithm becomes inefficient. This is where the probabilistic algorithm comes in. Instead of applying all maps to one point, this algorithm chooses randomly the map to apply and the resulting point is then drawn. Again it chooses another map randomly and is applied to the last point and so on. Because it only strores one point each time, this is very efficient. Moreover, it generates the image just as satisfactory. This algorithm is also known as the Chaos Game.

The fractals generated from IFS have true self-similarity. Their implementation is simple and most obey the matricial rule given by

Some examples include the following:

But other rules can also be used:

One of the most interesting features of IFS is the generation of 3D fractals. These fractals are characterized by zero volume but infinite area. One example is the Sierpinski piramid. Each side of the piramid is a Sierpinski triangle. A VRML version of this piramid is also available. You will have to save the file to your computer and then open from your computer. To visualize the VRML version you need Cosmoplayer that can be downloaded here.

Its fractal dimension is given by log(5)/log(2) = 2.3219.