Horseshoe map

The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator.

The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient.

The folding is done neatly, so that the orbits that remain forever in the square can be simply described.

Contracting by a factor smaller than one half assures that there will be a gap between the branches of the horseshoe.

Finally the resulting strip is folded into a horseshoe-shape and placed back into S. The interesting part of the dynamics is the image of the square into itself.

The extension of f to the caps adds a fixed point to the non-wandering set of the map.

To ensure that the map remains one-to-one, the contracted square must not overlap itself.

For example, the map on the right needs to be extended to a diffeomorphism of the sphere by using a “cap” that wraps around the equator.

The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit.

The set of points that never leaves the neighborhood of the given periodic orbit form a fractal.

Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in the left cap.

This is because the horseshoe maps the left cap into itself by an affine transformation that has exactly one fixed point.

That is, a point in Vn will, under n iterations of the horseshoe, end up in the set Hn of vertical strips.

If a point is to remain indefinitely in the square, then it must belong to a set Λ that maps to itself.

The structure of this set can be better understood by introducing a system of labels for all the intersections—a symbolic dynamics.