Math Conjectures

Furstenberg ×2,×3\times2,\times3 measure conjecture

OPENLandmarkConjectureProposed 1967 · Full conjecture

Canonical statement

Let μ\mu be a Borel probability measure on T=R/Z\mathbb T=\mathbb R/\mathbb Z that is invariant under T2(x)=2x(mod1)T_2(x)=2x\pmod1 and T3(x)=3x(mod1)T_3(x)=3x\pmod1, and ergodic for their joint action: every Borel set invariant modulo μ\mu under both maps has measure 00 or 11. Then either μ\mu is Lebesgue measure or there is a finite set F⊂TF\subset\mathbb T, invariant under both T2T_2 and T3T_3, such that μ(F)=1\mu(F)=1.
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Let \(\mu\) be a Borel probability measure on \(\mathbb T=\mathbb R/\mathbb Z\) that is invariant under \(T_2(x)=2x\pmod1\) and \(T_3(x)=3x\pmod1\), and ergodic for their joint action: every Borel set invariant modulo \(\mu\) under both maps has measure \(0\) or \(1\). Then either \(\mu\) is Lebesgue measure or there is a finite set \(F\subset\mathbb T\), invariant under both \(T_2\) and \(T_3\), such that \(\mu(F)=1\).
DYN-009:[Furstenberg1967DisjointnessErgodic][Rudolph1990Times2Times3][Johnson1992MeasuresInvariant][BurtonPanangaden2024FurstenbergFormulations]
Dynamical systemsProbability theoryNumber theory

Notes

Furstenberg's conjecture concerns Borel probability measures on the circle T\mathbb T that are simultaneously invariant and ergodic under doubling T2(x)=2xT_2(x)=2x and tripling T3(x)=3xT_3(x)=3x modulo one. It predicts extreme rigidity: any such measure is either Lebesgue measure or supported on a finite common orbit. The question grew out of Furstenberg's 1967 work on disjointness in ergodic theory, where the topological analogue — every infinite closed set invariant under both maps is all of T\mathbb T — was proved [Furstenberg1967DisjointnessErgodic].

The landmark partial result is due to Rudolph, who established the conclusion whenever the measure has positive entropy for one of the two maps [Rudolph1990Times2Times3]; Johnson extended this to measures invariant under a general nonlacunary multiplicative semigroup of integers [Johnson1992MeasuresInvariant]. The conjecture has also been recast in complex-analytic and operator-algebraic terms [BurtonPanangaden2024FurstenbergFormulations].

The nonatomic zero-entropy case remains completely open, and it is precisely there that the positive-entropy methods do not apply.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.