Rotation number

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

History

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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

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Suppose that is an orientation-preserving homeomorphism of the circle Then f may be lifted to a homeomorphism 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 Intuitively, it measures the average rotation angle along the orbits of f.

Example

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If is a rotation by (where ), then

and its rotation number is (cf. irrational rotation).

Properties

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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.

  • 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

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References

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  • Herman, Michael Robert (December 1979). "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations" [On the Differentiable Conjugation of Diffeomorphisms from the Circle to Rotations]. Publications Mathématiques de l'IHÉS (in French). 49: 5–233. doi:10.1007/BF02684798. S2CID 118356096., also SciSpace for smaller file size in pdf ver 1.3
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