Rotation number

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.

Suppose that $f:S^1\; \backslash to\; S^1$ is an orientation-preserving homeomorphism of the circle $S^1\; =\; \backslash R/\backslash Z.$ Then {{mvar|f}} 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 {{mvar|x}} and every integer {{mvar|m}}.

The rotation number of {{mvar|f}} is defined in terms of the iterates of {{mvar|F}}:

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

Henri Poincaré proved that the limit exists and is independent of the choice of the starting point {{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 of {{mvar|f}}.

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).

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

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

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. 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.

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

• {{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/ }}

• {{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