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 f may be lifted to a homeomorphism 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 2\pi N (where 0 < N < 1), then and its rotation number is N (cf. irrational rotation).

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 (not necessarily homeomorphic) 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. f–1 , but the limiting periodic orbits in forward and backward directions may be different. The rotation number is continuous when viewed as a map from the group of homeomorphisms (with C0 topology) of the circle into the circle.