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 f: S1 → S1 is an orientation preserving homeomorphism of the circle S1 = R/Z. Then f may be lifted to a homeomorphism F: R → R of the real line, satisfying
for every real number x and every integer m.
The rotation number of f is defined in terms of the iterates of F:
Henri Poincaré proved that the limit exists and is independent of the choice of the starting point x. The lift F is unique modulo integers, therefore the rotation number is a well-defined element of R/Z. Intuitively, it measures the average rotation angle along the orbits of f.

Example

If f is a rotation by 2πθ, then
then its rotation number is θ.

Properties

The rotation number is invariant under topological conjugacy, and even monotone topological semiconjugacy: if f and g are two homeomorphisms of the circle and
for a monotone continuous map h of the circle into itself then f and g 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 is continuous when viewed as a map from the group of homeomorphisms of the circle into the circle.