Standard Map

Here is a nice index pair for the standard map, with epsilon = 2.0. The green is all boxes in the tree, the black/cyan is the index pair, and the blue is the image of the index pair. The picture shows that the image of the index pair is completely contained within tree, so the result (topological entropy at least 0.2535) is definitely rigorous.

The following picture shows another index pair for epsilon = 2.0, produced by the "homoclinic insertion" method. It is followed by the same index pair translated by [0.5 0.5] to give a better picture of the fixed point region. Note how "skinny" the covering is (all boxes), which hints at the efficiency of the insertion algorithm. The entropy bound for this index pair is 0.3450.

Below is an index pair for epsion = 0.95, yielding an entropy bound of 0.1001. This was also produced by the homoclinic insertion method. The next picture is a region of this index pair showing homology on the second level (fortunately, these generators map to zero, so we do not have to deal with the alternating sum of traces).