Dynamical systems and ergodic theory Low-dimensional dynamics Theorem

Poincaré classification of circle homeomorphisms

Builds on Rotation number Homeomorphism Periodic point Semiconjugacy and factor map Topological conjugacy Irrational rotation Cantor set Dense set

Statement

Let $f$ be an orientation-preserving homeomorphism of the circle $S^1 = \R / \Z$ , with rotation number $\rho = \rho(f)$ .

Let $\rho = p/q$ with $p, q$ coprime integers and $q \ge 1$ . Then $f$ has a periodic point, and every periodic point of $f$ has least period $q$ . The omega-limit set of every point is a periodic orbit.
Let $\rho$ be irrational. Then $f$ has no periodic point. There is a continuous surjective map $h \colon S^1 \to S^1$ of degree one, weakly preserving the cyclic order, with $h \circ f = R_\rho \circ h$ . Here $R_\rho(x) = x + \rho$ .
If $\rho$ is irrational and $f$ has a dense orbit, then $h$ is a homeomorphism, so $f$ is topologically conjugate to the rotation $R_\rho$ .

Remarks

The converse parts also hold: $f$ has a periodic point if and only if $\rho$ is rational. For part 1, one shows that $\rho(F) = p/q$ for a lift $F$ forces the equation $F^q(x) = x + p$ to have a solution, which gives a periodic point. Since $f$ preserves the cyclic order, a periodic orbit of $f$ is arranged on the circle like an orbit of the rotation by $p/q$ . This forces its least period to be $q$ . Between two consecutive fixed points of $f^q$ the map $f^q$ moves every point in the same direction. So every orbit of $f^q$ is monotone and converges to a fixed point of $f^q$ . For part 2, $h$ is built by sending the orbit of a point $x$ to the orbit of $0$ under $R_\rho$ . This works because $f$ orders an orbit on the circle in the same way as $R_\rho$ does, and $h$ then extends by monotonicity. When $\rho$ is irrational and $f$ has no dense orbit, the omega-limit set of every point is the same Cantor set, and $f$ is only semiconjugate to $R_\rho$ . Example: for the rotation $R_{2/5}$ every point has period $5$ , and for an irrational $\alpha$ the map $R_\alpha$ is an irrational rotation with $h = \id$ .

Going deeper

For a student

The theorem reduces the topological study of circle homeomorphisms to one number plus a little extra structure. In the irrational case the remaining question is whether the semiconjugacy $h$ is a conjugacy, that is, whether $f$ has wandering intervals. Denjoy's theorem answers this for smooth enough diffeomorphisms: a $C^2$ diffeomorphism with irrational rotation number is conjugate to the rotation. In the rational case the periodic orbits all have the same period and the same combinatorics as the rotation by $p/q$ . This is the starting point of the study of Arnold tongues.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Borel sigma-algebra concept
Cantor set example
Complex exponential concept
Dense set concept
Equidistribution modulo one concept
Euclidean space object
Homeomorphism concept
Irrational number concept
Outer measure concept
Periodic point concept
Rotation number concept
Semiconjugacy and factor map concept
Topological conjugacy concept
Weak mixing concept
Equidistribution theorem statement
Lebesgue measure object
Irrational rotation example
Poincaré classification of circle homeomorphisms statement

Tag	`5N5SDY3K53`
URI	`m2://5N5SDY3K53`
Kind	statement, theorem
Writing	W0 stub
MSC	37E10, 37E45
Status	This copy of the site has no build data. Run `m2 status 5N5SDY3K53`.
About	Rotation number
Uses	Rotation number, Homeomorphism, Periodic point, Semiconjugacy and factor map, Topological conjugacy, Irrational rotation, Cantor set, Dense set