Rotation number

Short description: Invariant of homeomorphisms of the circle

In mathematics, the rotation number is an invariant of homeomorphisms of the circle.

## Contents

* 1 History
* 2 Definition
* 3 Example
* 4 Properties
* 5 See also
* 6 References
* 7 External links

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

[math]\displaystyle{ F(x + m) = F(x) +m }[/math]

for every real number x and every integer m.

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

[math]\displaystyle{ \omega(f)=\lim_{n\to\infty} \frac{F^n(x)-x}{n}. }[/math]

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

## Example

If [math]\displaystyle{ f }[/math] is a rotation by [math]\displaystyle{ 2\pi N }[/math] (where [math]\displaystyle{ 0 \lt N \lt 1 }[/math]), then

[math]\displaystyle{ F(x)=x+N, }[/math]

and its rotation number is [math]\displaystyle{ N }[/math] (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

[math]\displaystyle{ h\circ f = g\circ h }[/math]

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.

## See also

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

## References

* Herman, Michael Robert (December 1979). "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations" (in fr). Publications Mathématiques de l'IHÉS 49: 5–233. doi:10.1007/BF02684798. http://www.numdam.org/item/PMIHES_1979__49__5_0. , also SciSpace for smaller file size in pdf ver 1.3
* Sebastian van Strien, Rotation Numbers and Poincaré's Theorem (2001)[yes|permanent dead link|dead link}}]

## External links

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

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