This leaf-like image was produced by plotting the invariant set of the following four transformations:
S1(x, y) = (0, 0.1y)
S2(x, y) = (0.9x + 0.007y, 0.007x + 0.9y + 1.0)
S3(x, y) = (-0.24x + 0.28y - 0.014y2, 0.46x + 0.386y + 0.1 - 0.02y(10 - y) )
S4(x, y) = (0.24x - 0.28y + 0.015 y2, 0.46x + 0.386y + 0.1 - 0.02y(10 - y) )
These are all contractions on a subset of R2 on which the value of y is not too far from zero. It follows that there is a unique non-empty compact set that is invariant for these four transformations. (ref. Fractal Geometry by Falconer page 114).
If a set F is invariant for n transformations, Si with i = 1..n, it means that F is equal to the union of all the Si(F) for i = 1..n.
The same colour scheme was used as for the Barnsley fern, shown in the top left corner of the main page of this web site. The colour is set to dark green whenever a point is plotted as a result of the first transformation above being applied to another point in the invariant set, and faded towards a brighter shade when one of the other three transformations is used.