History and past results

Furstenburg himself never published an explicit statement of ``Furstenburg's Conjecture'', although he discussed it in lectures as a prototype of a more profound intuition about normal numbers. Recall that a real number is normal base $b$ if its $b$-adic expansion has the property that every finite block of digits has the expected frequency. Roughly speaking, this intuition says that it is very hard for a number to be abnormal with respect to two incommensurable bases.

The first explicit statement of Furstenburg's Conjecture occurs in the paper of Russell Lyons [ MR 89e:28031], who shows via elementary means that if $\mu$ is a measure on $\mathbb{T}$ that is jointly invariant under $\phi_p$ and $\phi_q$, and if $\phi_p$ is assumed to also have the strong property that it is a Kolmogorov automorphism of $(\mathbb{T},\mu)$ (i.e., that it has completely positive entropy), then $\mu$ must be Lebesgue measure.

The breakthrough came in 1990 with the paper of Dan Rudolph [ MR 91g:28026]. He showed using more sophisticated ergodic theory that if we merely assume that $\phi_p$ has positive entropy on $(\mathbb{T},\mu)$, then $\mu$ must be Lebesgue measure. Since then, several alternative proofs of Rudolph's theorem have been given, for example by Host [ MR 96g:11092] and Parry [ MR 97h:28009]. As the review of the latter states, `` It is striking that all the different approaches to the problem of the existence of non-Lebesgue, non-atomic, Borel measures invariant under $\phi_p$ and $\phi_q$ come up against the same entropy criterion.''

A 200-page account of the mathematical ideas surrounding Furstenburg's Conjecture and related topics by Klaus Schmidt is currently in draft form, entitled `` $\times2,\times3$, and $\times\beta$.''

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