Rotation number

 ( Fixed Points ) 

In mathematics, the rotation number is an invariant of 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 in terms of rational number of the rotation number.

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Definition Suppose that $f:S^1\; \backslash to\; S^1$ is an orientation-preserving homeomorphism of the circle group $S^1\; =\; \backslash R/\backslash Z.$ Then may be lifted to a homeomorphism $F:\; \backslash R\; \backslash to\; \backslash R$ of the real line, satisfying

$F(x\; +\; m)\; =\; F(x)\; +m$

for every real number and every integer .

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

$\backslash omega(f)=\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{F^n(x)-x\}\{n\}.$

Henri Poincaré proved that the limit exists and is independent of the choice of the starting point . The lift 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 .

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Example If $f$ is a rotation by $2\backslash pi\; N$ (where ), then

$F(x)=x+N,$

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

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Properties The rotation number is invariant under topological conjugacy, and even monotone topological semiconjugacy: if and are two homeomorphisms of the circle and

$h\backslash circ\; f\; =\; g\backslash circ\; h$

for a monotone continuous map of the circle into itself (not necessarily homeomorphic) then and 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 is a rational number (in the lowest terms). Then has a periodic orbit, every periodic orbit has period , and the order of the points on each such orbit coincides with the order of the points for a rotation by . Moreover, every forward orbit of converges to a periodic orbit. The same is true for backward orbits, corresponding to iterations of , but the limiting periodic orbits in forward and backward directions may be different.
• The rotation number of is an irrational number . Then has no periodic orbits (this follows immediately by considering a periodic point of ). There are two subcases.

# There exists a dense orbit. In this case is topologically conjugate to the irrational rotation by the angle and all orbits are dense set. Denjoy proved that this possibility is always realized when is twice continuously differentiable.

# There exists a Cantor set invariant under . Then is a unique minimal set and the orbits of all points both in forward and backward direction converge to . In this case, is semiconjugate to the irrational rotation by , and the semiconjugating map of degree 1 is constant on components of the complement of .

The rotation number is continuous when viewed as a map from the group of homeomorphisms (with topology) of the circle into the circle.

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See also

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

• , also SciSpace for smaller file size in pdf ver 1.3

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External links

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

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