Dynamical Systems and Ergodic Theory

This is the pattern of orbits produced by an area-preserving map of the torus. Each color is the orbit of a single point. The `phase plane' is divided up into a chaotic region and an ordered region (on which the orbits appear to be closed loops). Note the `islands' inside the ordered region in the middle of the portrait. This system has a parameter. For small values of the parameter, the ordered region appears to fill the phase plane, whereas for larger values of the parameter, the phase plane is almost all chaotic, with small ordered regions. It is unknown whether for some value of the parameter, the phase plane consists only of the chaotic part.

This `fern' is the attractor of an iterated function system. Three similarity transformations were chosen (a similarity transformation is a composition of a rotation, a scaling and a translation). One can see these transformations in the diagram: one sends the whole fern onto the bottom right leaf; one maps the fern to the bottom left leaf; and the final one maps scales the whole fern onto the smaller copy of the fern obtained by removing the bottom two leaves. A point was then chosen at random and a randomly chosen sequence of the maps was applied to the point. One can show that the `attractor' is independent of the initial point and is the same for `almost every' sequence of maps which are applied to the point.