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. […]