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
F \( {\displaystyle F(x)=x+\theta ,} \)
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.
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.
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
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)
Hellenica World - Scientific Library
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