Pinwheels and Quarter Turns

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ‎ denote the second iterate of the outer billiards map defined on R 2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ‎, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ‎. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.