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.

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.

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.

2. (ii) For any x, y ∈ ℝ with |x − y| ≤ k (k ∈ ℤ+) we have |T(x)| − T(y)| ≤ k.

3. […]

In this chapter we shall consider the stronger property of ergodicity for an invariant probability measure μ. This property is more appropriate (amongst other things) for understanding the “long term” average behaviour of a transformation.

Definitions and characterization of ergodic measures

Definition. Given a probability space (X, B, μ), a transformation T : X → X is called ergodic if for every set B ∈ B with T−1B = B we have that either μ(B) = 0 or μ(B) = 1.

Alternatively we say that μ is T-ergodic.

The following lemma gives a simple characterization in terms of functions.

Lemma 9.1. T is ergodic with respect to μ iff whenever f ∈ L1(X, B, μ) satisfies f = f ∘ T then f is a constant function.

Proof. This is an easy observation using indicator functions.

Poincaré recurrence and Kac's theorem

We begin with one of the most fundamental results in ergodic theory.

Theorem 9.2 (Poincaré recurrence theorem). Let T : X → X be a measurable transformation on a probability space (X, B, μ). Let A ∈ B have μ(A) > 0; then for almost points x ∈ A the orbit {Tnx}n ≥ 0 returns to A infinitely often.

Proof. Let F = {x ∈ A : Tnx ∉ A, ∀n ≥ 1}, then it suffices to show that μ(F) = 0.

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.).

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

1. (i) S : ℝ/ℤ → ℝ/ℤ defined by S(x) = 2x (mod 1), and

2. (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.