Horseshoe Map - The Horseshoe Map

The Horseshoe Map

The horseshoe map is a diffeomorphism defined from a region of the plane into itself. The region is a square capped by two semi-disks. The action of is defined through the composition of three geometrically defined transformations. First the square is contracted along the vertical direction by a factor . The caps are contracted so as to remain semi-disks attached to the resulting rectangle. Contracting by a factor smaller than one half assures that there will be a gap between the branches of the horseshoe. Next the rectangle is stretched horizontally by a factor of ; the caps remain unchanged. Finally the resulting strip is folded into a horseshoe-shape and placed back into .

The interesting part of the dynamics is the image of the square into itself. Once that part is defined, the map can be extended to a diffeomorphism by defining its action on the caps. The caps are made to contract and eventually map inside one of the caps (the left one in the figure). The extension of f to the caps adds a fixed point to the non-wandering set of the map. To keep the class of horseshoe maps simple, the curved region of the horseshoe should not map back into the square.

The horseshoe map is one-to-one, which means that an inverse f–1 exists when restricted to the image of S under f.

By folding the contracted and stretched square in different ways, other types of horseshoe maps are possible.

To ensure that the map remains one-to-one, the contracted square must not overlap itself. When the action on the square is extended to a diffeomorphism, the extension cannot always be done in the plane. 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 is an Axiom A diffeomorphism that serves as a model for the general behavior at a transverse homoclinic point, where the stable and unstable manifolds of a periodic point intersect.