Rotations

This opening chapter is devoted to the simplest dynamical systems on the circle that are not entirely trivial: the rigid rotations. Under a rigid rotation, all orbits look exactly the same. There are only two possible behaviours for such orbits. Either they are all dense on the circle, or else they are all periodic with the same period. This dichotomy can be read off from the angle by which points on the circle are rotated. The ratio of this angle to a full turn is called the rotation number.