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 15 - Invariant measures for commuting transformations
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 describe an important conjecture of Furstenberg and related work of Rudolph.
Furstenberg's conjecture and Rudolph's theorem
Consider the transformations
(i) S : ℝ/ℤ → ℝ/ℤ defined by S(x) = 2x (mod 1), and
(ii) T : ℝ/ℤ → ℝ/ℤ defined by T(x) = 3x (mod 1).
(For a mnemonic aid: S stands for “second” and T for “third”.) It is easy to see that these transformations commute, i.e. ST = TS).
Recall that the S-invariant probability measures form a convex weak-star compact set Ms (and similarly, the T-invariant probability measures form a convex weak-star compact set MT).
We want to describe the probability measures which are both T-invariant and S-invariant (i.e. the intersection Ms ∩ MT). We need only consider the (S, T)-ergodic measures μ in Ms ∩ MT (i.e. those probability measures invariant under both S and T for which the only Borel sets B with T−nS−mB = B ∀n, m ≥ 0 have either μ(B) = 0 or 1, since these are the extremal measures in Ms ∩ MT).
Furstenberg's conjecture. The only (S,T)-ergodic measures are the Haar-Lebesgue measure and measures supported on a finite set.
Notice that the Haar-Lebesgue measure v has entropies log 2 and log 3, respectively, for the transformations S and T, and any finitely supported measure always has zero entropy with respect to either S or T. The following partial solution is due to D.J. Rudolph.
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- Dynamical Systems and Ergodic Theory , pp. 153 - 160Publisher: Cambridge University PressPrint publication year: 1998