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 ${\displaystyle f:S^{1}\to S^{1))$ is an orientation-preserving homeomorphism of the circle ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} .}$ Then f may be lifted to a homeomorphism ${\displaystyle F:\mathbb {R} \to \mathbb {R} }$ of the real line, satisfying

${\displaystyle F(x+m)=F(x)+m}$

for every real number x and every integer m.

The rotation number of f is defined in terms of the iterates of F:

${\displaystyle \omega (f)=\lim _{n\to \infty }{\frac {F^{n}(x)-x}{n)).}$

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 ${\displaystyle \mathbb {R} /\mathbb {Z} .}$ Intuitively, it measures the average rotation angle along the orbits of f.

Example

If ${\displaystyle f}$ is a rotation by ${\displaystyle 2\pi N}$ (where ${\displaystyle 0<N<1}$), then

${\displaystyle F(x)=x+N,}$

and its rotation number is ${\displaystyle N}$ (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

${\displaystyle h\circ f=g\circ h}$

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.

• The rotation number of f is a rational number p/q (in the lowest terms). Then f has a periodic orbit, every periodic orbit has period q, and the order of the points on each such orbit coincides with the order of the points for a rotation by p/q. Moreover, every forward orbit of f converges to a periodic orbit. The same is true for backward orbits, corresponding to iterations of f ^–1, but the limiting periodic orbits in forward and backward directions may be different.
• The rotation number of f is an irrational number θ. Then f has no periodic orbits (this follows immediately by considering a periodic point x of f). There are two subcases.

1. There exists a dense orbit. In this case f is topologically conjugate to the irrational rotation by the angle θ and all orbits are dense. Denjoy proved that this possibility is always realized when f is twice continuously differentiable.
2. There exists a Cantor set C invariant under f. Then C is a unique minimal set and the orbits of all points both in forward and backward direction converge to C. In this case, f is semiconjugate to the irrational rotation by θ, and the semiconjugating map h of degree 1 is constant on components of the complement of C.

The rotation number is continuous when viewed as a map from the group of homeomorphisms (with C⁰ topology) of the circle into the circle.