The winding number
of a map
with initial value
is defined by
which represents the average increase in the angle per unit time (average frequency). A system with a rational winding number
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.