Furstenberg's Times 2, Times 3 Conjecture (a Short Survey)

TL;DR

This short survey analyzes Furstenberg's Times $2$, Times $3$ conjecture in the positive-entropy regime, summarizing three independent proofs (Rudolph, Host, Parry) and Lyons' earlier result. It highlights the central theme of measure rigidity for higher-rank actions on the circle, detailing Rudolph's symbolic-Z^2 approach, Parry's transfer-operator and Pinsker-algebra route, and Host's equidistribution mechanism for Host sequences. The results establish that, for coprime $p$ and $q$, atomless invariant measures with positive entropy under $ imes p$ and $ imes q$ must be Lebesgue, illustrating a broader rigidity phenomenon in ergodic theory. The survey situates these proofs in historical context, notes connections to Rudolph–Johnson extensions, and indicates how these ideas extend to related dynamical settings.

Abstract

The following is a concise exposition on the conjecture and three of its proofs for the case of positive entropy, by D. Rudolph [22] , by B. Host [14] and by W. Parry [21]. A simpler theorem of R. Lyons [19] - preceding them - is also presented and proved. This is a survey, no new results are introduced.