Lecture 13 - Rigid rotations of the circle
Today we move on from maps and flows in the plane and into maps and flows on the circle. Here again, we start with a relatively simply construction to help set the stage for more complicated stuff later on; Rigid circle rotations. Really, they come in only two types, depending on whether the rotation is rational or not (but even this takes a bit of definition.) But classifying circle rotations allows us to define what it means for an orbit to be dense in a space, and to define what it means for a point in the space to be recurrent with respect to a map. This leads to an important theorem whose statement actually defines a property; The Weyl Equidistribution Theorem. There are two applications here that are quite interesting. However, due to the timing of this tour, we will not cover them. Instead, we use circle rotations to begin the discussion of circle flows and toral maps and flows, where things get a little more subtle and complicated.