  We describe a 2 parameter family of polygon exchange transformations
parameterized by points in a square. Whenever the two parameters are
irrational, the polygon exchange has periodic orbits of arbitrarily large
period. We show that for almost all parameters, the polygon exchange map has
the property that almost every point is periodic. However, there is a dense set
of irrational parameters for which this fails. By choosing parameters
carefully, the measure of non-periodic points can be made arbitrarily close to
full measure. These results are powered by a notion of renormalization which
holds in a more general setting. Namely, we consider a renormalization of
tilings arising from the Corner Percolation Model.
