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 4 - Interval maps
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 concentrate on the special case of continuous maps on the closed interval I = [0, 1]. This level of specialization allows us to prove some particularly striking results on periodic points and topological entropy.
Fixed points and periodic points
Let T : I → I be a continuous map of the interval I = [0, 1] to itself. Recall that a fixed point x ∈ I satisfies Tx = x and that a periodic point (of period n) satisfies Tnx = x. We say that x has prime period n if n is the smallest positive integer with this property (i.e. Tkx ≠ x for k = 1,…, n – 1).
For interval maps a very simple visualization of fixed points exists. We can draw the graph GT of T : I → I and the diagonal D = {(x, x) : x ∈ I}.
Lemma 4.1. The fixed points Tx = x occur at the intersection points (x, x) ∈ GT ∩ D (see figure 4.1).
Similarly, if for n ≥ 2 we look for intersections of the graph (of n-compositions Tn : I → I) with the diagonal D then the intersection points (x, x) ∈ GT ∩ D are periodic points of period n.
Lemma 4.2. Assume that we have an interval J ⊂ I with T(J) ⊃ J; then there exists a fixed point Tx = x ∈ J.
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- Dynamical Systems and Ergodic Theory , pp. 33 - 46Publisher: Cambridge University PressPrint publication year: 1998