The Nature of the Compactification

This chapter looks more closely at Theorem 15.1 and gives more information about the PET that appears in that result. The basic idea of the proof is to remove from the torus Ŝ the singular set, i.e., the places where the PET is not defined. What is left over is isometric to the interior of a convex parallelotope. Section 16.2 analyzes the singular set and Section 16.3 constructs X1. Section 16.4 constructs the second parallelotope based on the action of the PET from Theorem 15.1. The proof of Theorem 16.1 finishes at the end of Section 16.4. Section 16.5 restates the case of Theorems 15.1 and 16.1 that apply to the pinwheel map associated to outer billiards on a polygon without parallel sides. The result is Theorem 16.9. Finally, Section 16.7 shows how Theorems 0.4 and 16.9 match up.