Rotation number


In mathematics, the rotation number is an invariant of homeomorphisms of the circle.

History

It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number.

Definition

Suppose that is an orientation-preserving homeomorphism of the circle Then may be lifted to a homeomorphism of the real line, satisfying
for every real number and every integer.
The rotation number of is defined in terms of the iterates of :
Henri Poincaré proved that the limit exists and is independent of the choice of the starting point. The lift is unique modulo integers, therefore the rotation number is a well-defined element of Intuitively, it measures the average rotation angle along the orbits of.

Example

If is a rotation by , then
and its rotation number is .

Properties

The rotation number is invariant under topological conjugacy, and even monotone topological semiconjugacy: if and are two homeomorphisms of the circle and
for a monotone continuous map of the circle into itself then and have the same rotation numbers. It was used by Poincaré and Arnaud Denjoy for topological classification of homeomorphisms of the circle. There are two distinct possibilities.
  • The rotation number of is a rational number . Then has a periodic orbit, every periodic orbit has period, and the order of the points on each such orbit coincides with the order of the points for a rotation by. Moreover, every forward orbit of converges to a periodic orbit. The same is true for backward orbits, corresponding to iterations of, but the limiting periodic orbits in forward and backward directions may be different.
  • The rotation number of is an irrational number. Then has no periodic orbits. There are two subcases.
The rotation number is continuous when viewed as a map from the group of homeomorphisms of the circle into the circle.