Furstenberg's Conjecture on ×2 and ×3 Invariant Sets

Let \(\times 2\) and \(\times 3\) act on the circle \(\mathbb{T} = \mathbb{R}/\mathbb{Z}\). Conjecture (Furstenberg, 1967): the only ergodic Borel probability measures invariant under both maps are Lebesgue measure and finitely supported (atomic) ones — equivalently, every infinite closed jointly invariant set is the whole circle. Rudolph (1990) proved it under positive entropy; the zero-entropy case is open.

Grew out of Furstenberg's 1967 disjointness theorem for ×2 and ×3 (times-two and times-three have no common non-trivial closed invariant sets beyond finite orbits of rationals). Rudolph and Johnson (1990-91) proved the measure-theoretic disjointness under positive entropy; Host, Lindenstrauss–Peres–Schlag and others extended rigidity to fractal dimensions. The pure topological classification above is where the wall sits.