Graph Master Picture Theorem

Abstract

This chapter aims to prove Theorem 0.4, the Graph Master Picture Theorem. Theorem 0.4 is proven in two different ways, the first proof is discussed here; it deduces Theorem 0.4 from Theorem 13.2, which is a restatement of [S1, Master Picture Theorem] with minor cosmetic changes. The chapter is organized as follows. Section 13.2 discusses the special outer billiards orbits on kites. Section 13.3 defines the arithmetic graph, which is an arithmetical way of encoding the behavior of a certain first return map of the special orbits. Section 13.4 states Theorem 13.2, a slightly modified and simplified version of [S1, Master Picture Theorem]. Section 13.5 deduces Theorem 0.4 from Theorem 13.2 and one extra piece of information. Finally, Section 13.6 lists the polytopes comprising the partition associated to Theorems 13.2 and 0.4.