Book contents
- Frontmatter
- Contents
- Introduction
- Preliminaries
- Chapter 1 Examples and basic properties
- Chapter 2 An application of recurrence to arithmetic progressions
- Chapter 3 Topological entropy
- Chapter 4 Interval maps
- Chapter 5 Hyperbolic toral automorphisms
- Chapter 6 Rotation numbers
- Chapter 7 Invariant measures
- Chapter 8 Measure theoretic entropy
- Chapter 9 Ergodic measures
- Chapter 10 Ergodic theorems
- Chapter 11 Mixing Properties
- Chapter 12 Statistical properties in ergodic theory
- Chapter 13 Fixed points for homeomorphisms of the annulus
- Chapter 14 The variational principle
- Chapter 15 Invariant measures for commuting transformations
- Chapter 16 Multiple recurrence and Szemeredi's theorem
- Index
Chapter 6 - Rotation numbers
Published online by Cambridge University Press: 05 March 2015
- Frontmatter
- Contents
- Introduction
- Preliminaries
- Chapter 1 Examples and basic properties
- Chapter 2 An application of recurrence to arithmetic progressions
- Chapter 3 Topological entropy
- Chapter 4 Interval maps
- Chapter 5 Hyperbolic toral automorphisms
- Chapter 6 Rotation numbers
- Chapter 7 Invariant measures
- Chapter 8 Measure theoretic entropy
- Chapter 9 Ergodic measures
- Chapter 10 Ergodic theorems
- Chapter 11 Mixing Properties
- Chapter 12 Statistical properties in ergodic theory
- Chapter 13 Fixed points for homeomorphisms of the annulus
- Chapter 14 The variational principle
- Chapter 15 Invariant measures for commuting transformations
- Chapter 16 Multiple recurrence and Szemeredi's theorem
- Index
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.
(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.
(ii) For any x, y ∈ ℝ with |x − y| ≤ k (k ∈ ℤ+) we have |T(x)| − T(y)| ≤ k.
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- Dynamical Systems and Ergodic Theory , pp. 57 - 64Publisher: Cambridge University PressPrint publication year: 1998