View source for Rotation number

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

In [[mathematics]], the '''rotation number''' is an [[Topological property|invariant]] of [[homeomorphism]]s of the [[circle]].

==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 orbit]]s in terms of [[rational number|rationality]] of the rotation number.

== Definition ==

Suppose that <math>f:S^1 \to S^1</math> is an orientation-preserving [[homeomorphism]] of the [[circle group|circle]] <math>S^1 = \R/\Z.</math> Then {{mvar|f}} may be [[Lift (mathematics)|lifted]] to a [[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 [[iterated function|iterates]] of {{mvar|F}}:

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

[[Henri Poincaré]] 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 [[orbit (dynamics)|orbits]] 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]]).

== Properties ==

The rotation number is invariant under [[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]] for topological classification of homeomorphisms of the circle. There are two distinct possibilities.

* The rotation number of {{mvar|f}} is a [[rational number]] {{mvar|p/q}} (in the lowest terms). Then {{mvar|f}} has a [[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]] {{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]] by the angle {{mvar|θ}} and all orbits are [[dense set|dense]]. Denjoy proved that this possibility is always realized when {{mvar|f}} is twice continuously differentiable.
:# There exists a [[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]]
* [[Denjoy's theorem on rotation number|Denjoy diffeomorphism]]
* [[Poincaré section]]
* [[Poincaré recurrence]]
* [[Poincaré–Bendixson 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]] |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]] | 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]]