Rotation number

Summary

The rotation number of an orientation-preserving homeomorphism of the circle is the average amount by which the map turns points around the circle, per step. For the rotation by an angle $\alpha$ it is $\alpha$. It is a topological invariant: homeomorphisms conjugate by an orientation-preserving homeomorphism have the same rotation number. Poincaré introduced it, and showed that it decides much of the dynamics: rational rotation number means periodic orbits, irrational rotation number means none. It is the basic invariant of circle dynamics and of quasi-periodic motion.