# Rotation number

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{{Short description|Invariant of homeomorphisms of the circle}}
{{distinguish|Rotation (quantity)|Rotation number (knot theory)}}
{{redirect-distinguish|Map winding number|Winding number|Turning number}}

In [mathematics](/source/mathematics), the '''rotation number''' is an [invariant](/source/Topological_property) of [homeomorphism](/source/homeomorphism)s of the [circle](/source/circle).

==History==

It was first defined by [Henri Poincaré](/source/Henri_Poincar%C3%A9) in 1885, in relation to the [precession](/source/precession) of the [perihelion](/source/perihelion) of a [planetary orbit](/source/planetary_orbit). Poincaré later proved a theorem characterizing the existence of [periodic orbit](/source/periodic_orbit)s in terms of [rationality](/source/rational_number) of the rotation number.

== Definition ==

Suppose that <math>f:S^1 \to S^1</math> is an orientation-preserving [homeomorphism](/source/homeomorphism) of the [circle](/source/circle_group) <math>S^1 = \R/\Z.</math> Then {{mvar|f}} may be [lifted](/source/Lift_(mathematics)) to a [homeomorphism](/source/homeomorphism) <math>F: \R \to \R</math> of the real line, satisfying

: <math> F(x + m) = F(x) +m </math>

for every real number {{mvar|x}} and every integer {{mvar|m}}.

The '''rotation number''' of {{mvar|f}} is defined in terms of the [iterates](/source/iterated_function) of {{mvar|F}}:

:<math>\omega(f)=\lim_{n\to\infty} \frac{F^n(x)-x}{n}.</math>

[Henri Poincaré](/source/Henri_Poincar%C3%A9) proved that the limit exists and is independent of the choice of the starting point {{mvar|x}}. The lift {{mvar|F}} is unique modulo integers, therefore the rotation number is a well-defined element of {{tmath|\R/\Z.}} Intuitively, it measures the average rotation angle along the [orbits](/source/orbit_(dynamics)) of {{mvar|f}}.

== Example ==

If <math>f</math> is a rotation by <math>2\pi N</math> (where <math>0 < N < 1</math>), then

: <math> F(x)=x+N, </math>

and its rotation number is <math>N</math> (cf. [irrational rotation](/source/irrational_rotation)).

== Properties ==

The rotation number is invariant under [topological conjugacy](/source/topological_conjugacy), and even monotone topological '''semiconjugacy''': if {{mvar|f}} and {{mvar|g}} are two homeomorphisms of the circle and

: <math> h\circ f = g\circ h </math>

for a monotone continuous map {{mvar|h}} of the circle into itself (not necessarily homeomorphic) then {{mvar|f}} and {{mvar|g}} have the same rotation numbers. It was used by Poincaré and [Arnaud Denjoy](/source/Arnaud_Denjoy) for topological classification of homeomorphisms of the circle. There are two distinct possibilities.

* The rotation number of {{mvar|f}} is a [rational number](/source/rational_number) {{mvar|p/q}} (in the lowest terms). Then {{mvar|f}} has a [periodic orbit](/source/periodic_orbit), every periodic orbit has period {{mvar|q}}, and the order of the points on each such orbit coincides with the order of the points for a rotation by {{mvar|p/q}}. Moreover, every forward orbit of {{mvar|f}} converges to a periodic orbit. The same is true for ''backward'' orbits, corresponding to iterations of {{math|''f''{{sup| –1}}}}, but the limiting periodic orbits in forward and backward directions may be different.
* The rotation number of {{mvar|f}} is an [irrational number](/source/irrational_number) {{mvar|θ}}. Then {{mvar|f}} has no periodic orbits (this follows immediately by considering a periodic point {{mvar|x}} of {{mvar|f}}). There are two subcases.

:# There exists a dense orbit. In this case {{mvar|f}} is topologically conjugate to the [irrational rotation](/source/irrational_rotation) by the angle {{mvar|θ}} and all orbits are [dense](/source/dense_set). Denjoy proved that this possibility is always realized when {{mvar|f}} is twice continuously differentiable.
:# There exists a [Cantor set](/source/Cantor_set) {{mvar|C}} invariant under {{mvar|f}}. Then {{mvar|C}} is a unique minimal set and the orbits of all points both in forward and backward direction converge to {{mvar|C}}. In this case, {{mvar|f}} is semiconjugate to the irrational rotation by {{mvar|θ}}, and the semiconjugating map {{mvar|h}} of degree 1 is constant on components of the complement of {{mvar|C}}.

The rotation number is ''continuous'' when viewed as a map from the group of homeomorphisms (with {{math|''C''{{sup|0}}}} topology) of the circle into the circle.

==See also==

* [Circle map](/source/Circle_map)
* [Denjoy diffeomorphism](/source/Denjoy's_theorem_on_rotation_number)
* [Poincaré section](/source/Poincar%C3%A9_section)
* [Poincaré recurrence](/source/Poincar%C3%A9_recurrence)
* [Poincaré–Bendixson theorem](/source/Poincar%C3%A9%E2%80%93Bendixson_theorem)

==References==

* {{cite journal |last=Herman |first=Michael Robert |title=Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations |journal=[Publications Mathématiques de l'IHÉS](/source/Publications_Math%C3%A9matiques_de_l'IH%C3%89S) |volume=49 |pages=5–233 |date=December 1979 |language=fr |trans-title=On the Differentiable Conjugation of Diffeomorphisms from the Circle to Rotations |url=http://www.numdam.org/item/PMIHES_1979__49__5_0 |doi=10.1007/BF02684798 |s2cid=118356096 }}, also ''SciSpace'' for smaller file size in [https://typeset.io/papers/sur-la-conjugaison-differentiable-des-diffeomorphismes-du-2klxn4vqsv pdf ver 1.3]

* {{cite journal| last=Poincaré | first=Henri | title= Sur les courbes définies par les équations différentielles (III) |language=fr| journal=
[Journal de Mathématiques Pures et Appliquées](/source/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es) | date= 1885 | volume =1 |pages= 167-244 | url= http://www.numdam.org/item/JMPA_1885_4_1__167_0/ }}

== External links ==
* {{Scholarpedia|title=Rotation theory|urlname=Rotation_theory|curator=Michał Misiurewicz}}
* Weisstein, Eric W. [http://mathworld.wolfram.com/MapWindingNumber.html "Map Winding Number"]. From MathWorld--A Wolfram Web Resource.

Category:Fixed points (mathematics)
Category:Dynamical systems
Category:Rotation

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Adapted from the Wikipedia article [Rotation number](https://en.wikipedia.org/wiki/Rotation_number) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rotation_number?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.