Monotone Maps of the Circle

Abstract

Throughout this monograph the following conventions are adopted:

• The circle is represented as the quotient ${\mathbb T} = {\mathbb R}/{\mathbb Z}$.

• $\pi : {\mathbb R} \to {\mathbb T}$ is the canonical projection.

• Three or more distinct points $t_1, t_2, \ldots , t_k \in {\mathbb T}$ are in positive cyclic order if there are representatives x _i  ∈ π ⁻¹(t _i ) such that x ₁ < x ₂ < ⋯ < x _k  < x ₁ + 1.

• For a distinct pair $t_1,t_2 \in {\mathbb T}$, the interval $(t_1,t_2) \subset {\mathbb T}$ is defined as the set of all $t \in {\mathbb T}$ such that t ₁, t, t ₂ are in positive cyclic order. We define the intervals (t ₁, t ₂], [t ₁, t ₂), [t ₁, t ₂] by adding the suitable endpoints to (t ₁, t ₂).

• The length of an interval $(t_1,t_2) \subset {\mathbb T}$ is always understood as its normalized Lebesgue measure, that is, the unique representative of t ₂ − t ₁ in [0, 1).