Winding Number (Map)

The winding number of a map is defined by

$$W \equiv\lim_{n\to\infty}{f^n(\theta)-\theta\over n},$$

which represents the average increase in the angle $\theta$ per unit time (average frequency). A system with a Rational winding number $W = {p/q}$ is Mode-Locked, whereas a system with an Irrational winding number is Quasiperiodic. Note that since the Rationals are a set of zero Measure on any finite interval, almost all winding numbers will be irrational, so almost all maps will be Quasiperiodic.