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

$F(x+m)=F(x)+m$

for every real number Template:Mvar and every integer Template:Mvar.

The rotation number of Template:Mvar is defined in terms of the iterates of Template:Mvar:

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

Henri Poincaré proved that the limit exists and is independent of the choice of the starting point Template:Mvar. The lift Template:Mvar is unique modulo integers, therefore the rotation number is a well-defined element of Template:Tmath Intuitively, it measures the average rotation angle along the orbits of Template:Mvar.

Example

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

$F(x)=x+N,$

and its rotation number is $N$ (cf. irrational rotation).

Properties

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

$h\circ f=g\circ h$

for a monotone continuous map Template:Mvar of the circle into itself (not necessarily homeomorphic) then Template:Mvar and Template:Mvar 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 Template:Mvar is a rational number Template:Mvar (in the lowest terms). Then Template:Mvar has a periodic orbit, every periodic orbit has period Template:Mvar, and the order of the points on each such orbit coincides with the order of the points for a rotation by Template:Mvar. Moreover, every forward orbit of Template:Mvar 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 Template:Mvar is an irrational number Template:Mvar. Then Template:Mvar has no periodic orbits (this follows immediately by considering a periodic point Template:Mvar of Template:Mvar). There are two subcases.

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

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

See also

• Circle map
• Denjoy diffeomorphism
• Poincaré section
• Poincaré recurrence
• Poincaré–Bendixson theorem

References

• , also SciSpace for smaller file size in pdf ver 1.3

External links

• Weisstein, Eric W. "Map Winding Number". From MathWorld--A Wolfram Web Resource.