The horseshoe map is motivated by studying the dynamics of a map in the vicinity of a periodic orbit with a homoclinic point. Such a system gives rise to a homoclinic tangle. Consider a small box (ABCD) in the vicinity of a periodic orbit as seen from the surface of section. This situation is illustrated in Figure 4.18(a). The box is chosen so that the side AD is part of the unstable manifold, and the sides AB and DC are part of the stable manifold. We now ask how this box evolves under forward and backward iterations. The unstable and stable manifolds of the periodic orbit are invariant. Therefore, when the box is iterated, any point of the box that lies on an invariant manifold must always remain on this invariant manifold.
Figure 4.18: Formation of a horseshoe inside a homoclinic tangle.
If we iterate points in the box forward, then we generally end up (after a
finite number of iterations) with the ``horseshoe
shape'' (C'D'A'B') shown in Figure
4.18(b). The initial segment AD, which lies on the unstable manifold,
gets mapped to the segment A'D', which is also part of the unstable manifold.
Similarly, if we iterate the box backward we find a backward horseshoe
perpendicular to the forward horseshoe (Fig. 4.18(c)). The box of
initial points gets compressed along the unstable manifold 
and stretched
along the stable manifold 
. After a finite number of iterations, the
forward image of the box will intersect the backward image of the box.
Further iteration produces more intersections.
Each new region of intersection contains a periodic orbit (see Fig. 4.20) as well as segments of the unstable and stable manifolds. That is, the horseshoe can be viewed as generating the homoclinic tangle. Smale realized that this type of horseshoe structure occurs quite generally in a chaotic system. Therefore, he decided to isolate this horseshoe map from the rest of the problem [11].
A schematic for this isolated horseshoe map is presented in Figure 4.18(d). Like the quadratic map, it consists of a stretch and a fold. The horseshoe map can be thought of as a ``thickened'' quadratic map. Unlike the quadratic map, though, the horseshoe map is invertible. The future and past of all points are well defined. However, the itinerary of points that get mapped out of the box are ignored. We are only concerned with points that remain in the box under all future and past iterations. These points form the invariant set.