Chapter 6 - Rotation numbers

Summary

In this chapter we shall define the useful concept of the rotation number for orientation preserving homeomorphisms of the circle.

Homeomorphisms of the circle and rotation numbers

Let T : ℝ/ℤ → ℝ/ℤ be an orientation preserving homeomorphism of the circle to itself. There is a canonical projection π : ℝ → ℝ/ℤ given by π(x) = x (mod 1). We call a monotone map T : ℝ → ℝ a lift of T if the canonical projection π : ℝ → ℝ/ℤ is a semi-conjugacy (i.e. π ∘ T = T ∘ π).

For a given map T : ℝ/ℤ → ℝ/ℤ a lift T : ℝ → ℝ will not be unique.

Example. If T(x) = (x + α) (mod 1) then for any k ∈ ℤ the map T : ℝ → ℝ defined by T(x) = x + α + k is a lift. To see this observe that π(T(x)) = π(x + α + k) = x + α (mod 1) and T(π(x)) = π(x) + α(mod 1) = x + α (mod 1).

The following lemma summarizes some simple properties of lifts.

Lemma 6.1.

1. (i) Let T : ℝ/ℤ → ℝ/ℤ be a homeomorphism of the circle; then if T : ℝ → ℝ is a lift, then any other lift T′ : ℝ → ℝ must be of the form T′(x) = T(x) + k, for some k ∈ ℤ.

2. (ii) For any x, y ∈ ℝ with |x − y| ≤ k (k ∈ ℤ⁺) we have |T(x)| − T(y)| ≤ k.

3. […]