From Horseshoes to Tangles

  Forward and backward iterations of the horseshoe map generate the locations of periodic points to a higher and higher precision. That is, by iterating the horseshoe map, we can specify the location of a periodic orbit within a homoclinic tangle (of the horseshoe) to any degree of accuracy. For instance, after one iteration, we know the approximate location of a period two orbit. It lies somewhere within the shaded regions labeled 0.1 and 1.0 in Figure 4.20(c). After two iterations, we know its position even better. It lies somewhere within the shaded regions labeled 10.10 and 01.01 (Fig.\ 4.20(c)).

The forward iterates of the horseshoe map produce a ``snake'' that approaches the unstable manifold of the periodic point. The backward iterates produce another snake that approaches the stable manifold of the periodic point at the origin. Thus, iterating a horseshoe generates a tangle. The relative locations of horizontal and vertical branches of this tangle are the same as those that occur in a homoclinic tangle with a horseshoe arising in a particular flow.

  
Figure 4.22: Homoclinic (horseshoe) tangle and the labeling scheme for horizontal and vertical branches from a pair of alternating binary trees.

This is illustrated in Figure 4.22. We can name the branches of the tangle with the same labeling scheme we used for the horseshoe. For a horseshoe, the labeling scheme is easy to see once we notice that both the horizontal and vertical branches are labeled according to the alternating binary tree introduced in section 2.12.2.

The labeling of the horizontal branches is determined by the symbols For instance, the horizontal label for branch can be determined by reading down the alternating binary tree as illustrated in Figure 4.22. A second alternating binary tree is used to determine the labeling for the vertical branches. The labeling for the branch is indicated in Figure 4.22. The labeling scheme for the horizontal and vertical branches at first appears complicated. However, the branch names are easy to write down once we realize that they can be read directly from the alternating binary tree.