Statement of the Furstenberg Conjecture

Furstenberg's Conjecture, in its most concrete form, is the following. Let $\mathbb{T}$ denote the additive circle group, and for each integer $m>1$ let $\phi_m\colon\mathbb{T}\to\mathbb{T}$ be defined by $\phi_m(t)=mt\pmod1$. Then the only atomless probability measure on $\mathbb{T}$ that is simultaneously invariant under both $\phi_2$ and $\phi_3$ is Haar measure.

To say that a measure $\mu$ is invariant under a map $\phi$ means that $\mu(\phi^{-1}(E))=\mu(E)$ for all measureable sets $E$.

A more general version of this conjecture raises the same question for the maps $\phi_p$ and $\phi_q$, where $p$ and $q$ are positive integers satisfying the necessary condition that no power of $p$ equals a power of $q$, i.e. that $\log p$ and $\log q$ are rationally independent.

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