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
Preliminaries
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
1. Conventions. The book is divided into 16 chapters, each subdivided into sections numbered in order (e.g. chapter 12, section 3 is numbered 12.3).
Within each chapter results (Theorems, Propositions or Lemmas) are labelled by the chapter and then the order of occurrence (e.g. the fifth result in chapter 3 is Proposition 3.5). The exceptions to this rule are: sublemmas which are presented within the context of the proof of a more important result (e.g. the proof of Theorem 2.2 contains Sublemmas 2.2.1 and 2.2.2); and corollaries (the corollary to Theorem 5.5 is Corollary 5.5.1).
We denote the end of a proof by ▪.
Finally, equations are numbered by the chapter and their order of occurrence (e.g. the fourth equation in chapter 5 is labelled (5.4))
2. Notation. We shall use the standard notation: ℝ to denote the real numbers; ℚ to denote the rational numbers; ℤ to denote the integer numbers; ℕ to denote the natural numbers; and ℤ+ to denote the nonnegative integers. We use the convenient convention that: ℝ/ℤ = {x + ℤ : x ∈ ℝ} (which is homeomorphic to the standard unit circle); ℝ2/ℤ2 = {(x1, x2) + ℤ2 : (x1, x2) ∈ ℝ2} (which is homeomorphic to the standard 2-torus); etc. However, for x ∈ ℝ we denote the corresponding element in ℝ/ℤ by x (mod 1) (and similarly for ℝ2/ℤ2, etc.).
- Type
- Chapter
- Information
- Dynamical Systems and Ergodic Theory , pp. xi - xivPublisher: Cambridge University PressPrint publication year: 1998