Rotation Number and Circle Maps — Mathematics Knowledge System

Overview

The rotation number rho(f) of a circle homeomorphism f: S^1 -> S^1 is a real number that measures the average amount by which f rotates the circle per iteration. It is defined as rho(f) = lim_{n->inf} (F^n(x) - x)/n for any x, where F: R -> R is a lift of f to the real line. Poincare's theorem: rho(f) is rational p/q iff f has a periodic orbit of period q; rho(f) is irrational iff all orbits are dense (Denjoy theorem for diffeomorphisms). The rotation number is a complete topological invariant for irrational circle rotations.

Intuition

Imagine a circle being stretched and rotated. If after q iterations every point returns to its starting position, the rotation number is p/q (the map makes p full rotations in q steps). If the rotation number is irrational (like sqrt(2) - 1), no point ever returns -- orbits are dense on the circle, filling it uniformly. The rotation number is like the 'average angle' added per iteration, measured in full rotations.

Formal Definition

Definition

A circle homeomorphism f: R/Z -> R/Z lifts to F: R -> R with F(x+1) = F(x) + 1 and pi o F = f o pi (pi: R -> R/Z). The rotation number rho(f) = lim_{n->inf} (F^n(x) - x)/n exists and is independent of x (for homeomorphisms). It is independent of the choice of lift modulo integers. Poincare's theorem: (1) rho(f) = p/q (rational) iff f has a periodic orbit of period q; (2) rho(f) is irrational iff f is semi-conjugate to the rigid rotation R_{rho(f)}: x -> x + rho(f). Denjoy: if f is C^2 and rho is irrational, f is topologically conjugate (not just semi-conjugate) to R_{rho}.

\rho(f) = \lim_{n\to\infty} \frac{F^n(x) - x}{n}

Rotation number definition

\rho(f) = p/q \iff f \text{ has a periodic orbit of period } q

Poincare's rationality theorem

\rho(f) \notin \mathbb{Q} \Rightarrow f \text{ is minimal (all orbits dense)}

Denjoy minimality

Notation

Notation	Meaning
$\rho(f)$	Rotation number of circle homeomorphism f
$R_\alpha: x \mapsto x + \alpha$	Rigid rotation by alpha
$F$	Lift of f to R (F(x+1) = F(x)+1)

Theorems

Theorem 1: Poincare's Theorem on Rotation Number

For a circle homeomorphism f with lift F: (1) The limit rho(f) = lim(F^n(x)-x)/n exists for all x and is independent of x. (2) rho(f) is rational p/q (in lowest terms) iff f has a periodic orbit of least period q. (3) rho(f) is irrational iff f is semi-conjugate to the rotation R_{rho(f)} and all orbits are dense.

Theorem 2: Denjoy's Theorem

If f: S^1 -> S^1 is a C^2 orientation-preserving diffeomorphism with irrational rotation number rho, then f is topologically conjugate to the rigid rotation R_rho. Concretely: there exists a homeomorphism h: S^1 -> S^1 with h o f = R_rho o h. The C^2 hypothesis is sharp: Denjoy constructed C^1 examples with irrational rotation number that are NOT minimal.

Worked Examples

1
The lift is F(x) = x + alpha (with no modular reduction).
2
F^n(x) = x + n*alpha.
3
Rotation number: rho(f) = lim(F^n(x)-x)/n = lim(n*alpha)/n = alpha.
$\rho(f) = \lim_{n\to\infty}\frac{x + n\alpha - x}{n} = \alpha$
4
If alpha is irrational (e.g., sqrt(2)-1), all orbits {x + n*alpha mod 1} are dense on S^1. If alpha = p/q, every orbit has period q.

✓ Answer

The rotation number of R_alpha is alpha itself. Rational alpha -> periodic orbits; irrational -> dense orbits.

Practice Problems

Mediumfree response

What is the 'Devil's staircase' in the context of rotation numbers, and why is it called that?

Common Mistakes

Common Mistake

Confusing the rotation number with the average rotation in degrees.

The rotation number is measured in FULL ROTATIONS (not degrees or radians). rho = 1/4 means the map rotates by 1/4 of a full turn (90 degrees) on average per iterate. rho = 1 would mean one full rotation, which is the same as rho = 0 on the circle (rotation number is defined mod 1 for orientable maps, but the value in [0,1) is canonical). Use rho in [0,1) to avoid ambiguity.

Quiz

A circle homeomorphism f has rotation number rho(f) = 2/3. This implies:

Every orbit of f is dense on S^1f has a periodic orbit of period 3f is conjugate to an irrational rotationf has no fixed points

Historical Background

Poincare introduced the rotation number in his 1885 work on differential equations on surfaces. He proved the rationality criterion for periodic orbits. Denjoy (1932) proved that a C^2 circle diffeomorphism with irrational rotation number is topologically conjugate to an irrational rotation. The Devil's staircase (Cantor function) arises as the rotation number as a function of a parameter in the family of 'standard maps' -- it is constant on rationals and strictly increasing only on irrationals. The Arnold tongues organize the parameter space.

1885
Poincare introduces rotation number for circle maps
Henri Poincare
1932
Denjoy proves C^2 diffeomorphisms with irrational rotation number are minimal
Arnaud Denjoy
1961
Arnold studies the standard circle map and Arnold tongues
Vladimir Arnold

Summary

Rotation number: rho(f) = lim(F^n(x)-x)/n. Average rotation per iterate, independent of x.
Rational rho = p/q: periodic orbits of period q. Irrational: all orbits dense (Denjoy).
Denjoy theorem: C^2 circle diffeomorphism with irrational rho is conjugate to rigid rotation R_rho.
Devil's staircase: rotation number as function of parameter is Cantor-function-shaped.

References

BookDevaney, R.L. An Introduction to Chaotic Dynamical Systems. 2nd ed. Addison-Wesley, 1989.
WebsiteWikipedia -- Rotation number