Rotation number

This article is about the rotation number, which is sometimes called the map winding number or simply winding number. There is another meaning for winding number, which appears in complex analysis.

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.

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Definition

Suppose that f: S1S1 is an orientation preserving homeomorphism of the circle S1 = R/Z. Then f may be lifted to a homeomorphism F: RR of the real line, satisfying

 F(x %2B m) = F(x) %2Bm

for every real number x and every integer m.

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

\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 R/Z. Intuitively, it measures the average rotation angle along the orbits of f.

Example

If f is a rotation by θ, so that

 F(x)=x%2B\theta,

then its rotation number is θ (cf Irrational rotation).

Properties

The rotation number is invariant under topological conjugacy, and even topological semiconjugacy: if f and g are two homeomorphisms of the circle and

 h\circ f = g\circ h

for a 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.

  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.

Rotation number is continuous when viewed as a map from the group of homeomorphisms (with  C^0 topology) of the circle into the circle.

See also

References