Existence of Infinite Orbits

Abstract

This chapter begins Part 5 of the book. This part is devoted mostly to the study of the distribution of the plaid polygons: their size and number depending on the parameter. Section 21.2 gives a criterion for a point in FX to have a well-defined orbit. Section 21.3 revisits the pixelated spacetime diagrams of capacity 2, and uses them to construct a large supply of plaid polygons having large diameter. The construction in Section 21.3 works one parameter at a time. Section 21.4 takes the limit of our construction relative to a sequence of even rational parameters converging to our irrational parameter. This limiting argument completes the proof. Section 21.5 explains how to associate a plaid path to an infinite orbit. Section 21.6 gives a quick alternate proof of Theorem 21.1, based on results from [S1].