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.