I will begin by describing a method to renormalize a dynamical system associated with a class of tilings in the plane related to corner percolation studied by Gábor Pete. I will explain how these ideas give rise to a renormalization scheme for a 2-parameter family of piecewise isometries of the square pillowcase. I'll describe some results about the dynamics of these maps. Periodic points are topologically generic for all these maps, so it is natural to study the aperiodic points.