Math 2.0

Dynamical systems and ergodic theory Topological dynamics Theorem

Poincaré–Birkhoff theorem

Builds on AnnulusHomeomorphismLebesgue measureUniversal cover

Statement

Let A=(ℝ/ℤ)×[0,1]A = (\R / \Z) \times [0, 1] be the closed annulus, with the area measure dθ drd\theta \, dr given by Lebesgue measure. Let f ⁣:A→Af \colon A \to A be a homeomorphism such that:

  1. ff preserves area.
  2. ff maps each boundary circle (ℝ/ℤ)×{0}(\R / \Z) \times \set{0} and (ℝ/ℤ)×{1}(\R / \Z) \times \set{1} to itself.
  3. ff twists the boundary circles in opposite directions: some lift F ⁣:ℝ×[0,1]→ℝ×[0,1]F \colon \R \times [0, 1] \to \R \times [0, 1] of ff to the universal cover, written F(x,r)=(x+g(x,r),h(x,r))F(x, r) = (x + g(x, r), h(x, r)), satisfies g(x,0)<0<g(x,1)g(x, 0) < 0 < g(x, 1) for all x∈ℝx \in \R.

Then ff has at least two fixed points.

Remarks

Poincaré stated the theorem shortly before his death and checked many special cases, and Birkhoff published a proof in 1913. Birkhoff's proof works with the lift FF on the strip ℝ×[0,1]\R \times [0, 1]. The twist condition makes the horizontal displacement gg change sign between the two boundaries. Area preservation then rules out the ways in which FF could still avoid a fixed point. Applied to iterates fqf^q and lifts shifted by integers pp, the theorem gives periodic orbits of every rotation number p/qp/q between the rotation numbers of the two boundaries. Area preservation is necessary: the map with lift F(x,r)=(x+r−1/2,r2)F(x, r) = (x + r - 1/2, r^2) satisfies 2 and 3, but a fixed point would need r2=rr^2 = r and r−1/2∈ℤr - 1/2 \in \Z, which is impossible. This map is not area preserving, since its Jacobian determinant is 2r2r.

Going deeper

For a student

Area-preserving twist maps of an annulus arise as return maps of Hamiltonian systems with two degrees of freedom, for example near an elliptic periodic orbit or in billiards. The theorem then produces infinitely many periodic orbits, one family for each rational rotation number in the twist interval. It is a forerunner of symplectic topology: Arnold's conjecture on fixed points of Hamiltonian diffeomorphisms was proposed as a higher-dimensional generalisation.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Annulus, concept, no evidence levelAnnulusconceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHomeomorphism, concept, no evidence levelHomeomorphismconceptOuter measure, concept, no evidence levelOuter measureconceptUniversal cover, concept, no evidence levelUniversal coverconceptLebesgue measure, object, no evidence levelLebesgue measureobjectPoincaré–Birkhoff theorem, statement, no evidence levelPoincaré–Birkhoff the…statement
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.
  • no evidence level
  1. Annulus concept
  2. Borel sigma-algebra concept
  3. Euclidean space object
  4. Homeomorphism concept
  5. Outer measure concept
  6. Universal cover concept
  7. Lebesgue measure object
  8. Poincaré–Birkhoff theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff_theorem
About this page
TagK7MR77CHP6
URIm2://K7MR77CHP6
Kindstatement, theorem
WritingW0 stub
MSC37E40, 37E30, 37J12
StatusThis copy of the site has no build data. Run m2 status K7MR77CHP6.
AboutAnnulus
UsesAnnulus, Homeomorphism, Lebesgue measure, Universal cover
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.