In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
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.
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
Intuitively, it measures the average rotation angle along the orbits of f. If
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.