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: S¹ → S¹ is an orientation-preserving homeomorphism of the circle S¹ = R/Z. Then f may be lifted to a homeomorphism F: R → 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 R/Z. Intuitively, it measures the average rotation angle along the orbits of f.

Example[]

If f is a rotation by 2πθ (where 0≤θ<1), then

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

then its rotation number is θ (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 ${\displaystyle C^{0}}$ topology) of the circle into the circle.

See also[]

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

References[]

• M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979) pp. 5–234
• Sebastian van Strien, Rotation Numbers and Poincaré's Theorem (2001)

External links[]

• Michał Misiurewicz (ed.). "Rotation theory". Scholarpedia.
• Weisstein, Eric W. "Map Winding Number." From MathWorld--A Wolfram Web Resource