Rotation number

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

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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: S1S1 is an orientation preserving homeomorphism of the circle S1 = R/Z. Then f may be lifted to a homeomorphism F: RR of the real line, satisfying

for every real number x and every integer m.

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

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

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

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.

  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 topology) of the circle into the circle.

See also

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References