Rotation number explained

Rotation number should not be confused with Rotation (quantity).

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} = \R/\Z. Then may be lifted to a homeomorphism F: \R \to \R of the real line, satisfyingF(x + m) = F(x) + m

for every real number and every integer .

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

\omega(f) = \lim _{n\toinfty} (F ^{n(x)-x})/(n) .

Henri Poincaré proved that the limit exists and is independent of the choice of the starting point . The lift 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 .

Example

If

f is a rotation by 2 \pi N (where 0 < N < 1 ), thenF(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 and are two homeomorphisms of the circle and

h\circ f = g\circ h

for a monotone continuous map of the circle into itself (not necessarily homeomorphic) then and 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 is a rational number (in the lowest terms). Then has a periodic orbit, every periodic orbit has period, and the order of the points on each such orbit coincides with the order of the points for a rotation by . Moreover, every forward orbit of converges to a periodic orbit. The same is true for backward orbits, corresponding to iterations of, but the limiting periodic orbits in forward and backward directions may be different.
• The rotation number of is an irrational number . Then has no periodic orbits (this follows immediately by considering a periodic point of). There are two subcases.

1. There exists a dense orbit. In this case is topologically conjugate to the irrational rotation by the angle and all orbits are dense. Denjoy proved that this possibility is always realized when is twice continuously differentiable.

1. There exists a Cantor set invariant under . Then is a unique minimal set and the orbits of all points both in forward and backward direction converge to . In this case, is semiconjugate to the irrational rotation by, and the semiconjugating map of degree 1 is constant on components of the complement of .

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

See also

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

References

• Herman . Michael Robert . Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations . . 49 . 5–233 . December 1979 . fr . On the Differentiable Conjugation of Diffeomorphisms from the Circle to Rotations . 10.1007/BF02684798 . 118356096 ., also SciSpace for smaller file size in pdf ver 1.3
• Poincaré . Henri . Sur les courbes définies par les équations différentielles (III) . fr. . 1885 . 1 . 167-244 .

External links

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