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| :''This article is about the ''rotation number'', which is sometimes called the ''map winding number'' or simply ''winding number''. There is another meaning for [[winding number]], which appears in [[complex analysis]].''
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| In [[mathematics]], the '''rotation number''' is an [[Topological property|invariant]] of [[homeomorphism]]s of the [[circle]]. 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 orbit]]s in terms of [[rational number|rationality]] of the rotation number.
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| == Definition == | |
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| Suppose that ''f'': ''S''<sup>1</sup> → ''S''<sup>1</sup> is an orientation preserving [[homeomorphism]] of the [[circle]] ''S''<sup>1</sup> = [[Circle group|'''R'''/'''Z''']]. Then ''f'' may be [[Lift (mathematics)|lifted]] to a [[homeomorphism]] ''F'': '''R''' → '''R''' of the real line, satisfying
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| : <math> F(x + m) = F(x) +m </math>
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| for every real number ''x'' and every integer ''m''.
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| The '''rotation number''' of ''f'' is defined in terms of the [[iterated function|iterates]] of ''F'':
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| :<math>\omega(f)=\lim_{n\to\infty} \frac{F^n(x)-x}{n}.</math>
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| [[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 [[orbit (dynamics)|orbits]] of ''f''.
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| === Example ===
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| If ''f'' is a rotation by ''θ'', so that
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| : <math> F(x)=x+\theta, </math>
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| then its rotation number is ''θ'' (cf [[Irrational rotation]]).
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| == Properties ==
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| The rotation number is invariant under [[topological conjugacy]], and even topological '''semiconjugacy''': if ''f'' and ''g'' are two homeomorphisms of the circle and
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| : <math> h\circ f = g\circ h </math>
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| for a 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.
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| * 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''<sup>−1</sup>, but the limiting periodic orbits in forward and backward directions may be different.
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| * 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.
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| :# 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 set|dense]]. Denjoy proved that this possibility is always realized when ''f'' is twice continuously differentiable.
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| :# 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''. | |
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| Rotation number is ''continuous'' when viewed as a map from the group of homeomorphisms (with <math> C^0 </math> topology) of the circle into the circle.
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| ==See also==
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| * [[Circle map]]
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| * [[Denjoy diffeomorphism]]
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| * [[Poincaré section]]
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| * [[Poincaré recurrence]]
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| ==References==
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| * M.R. Herman, ''Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations'', Publ. Math. IHES, 49 (1979) pp. 5–234
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| * {{Scholarpedia|title=Rotation theory|urlname=Rotation_theory|curator=Michał Misiurewicz}}
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| * Sebastian van Strien, ''[http://www.maths.warwick.ac.uk/~strien/MA424/HTMLversion/node6.html Rotation Numbers and Poincaré's Theorem]'' (2001)
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Dynamical systems]]
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