• (The problem of smooth realizations) Let $X$ be a Lebesgue space with measure $\mu$, and let $T:X\to X$ be a transformation preserving the measure $\mu$. If the entropy $h_\mu(T)$ is finite, is $(X,T,\mu)$ always measurably isomorphic to a smooth system $(M,f,v)$, where $M$ is a compact manifold, $f$ is a diffeomorphism of $M$ and $v$ is a smooth volume?
• (Furstenberg's $\times 2 \times 3$ problem) Does there exist a Borel probability measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$, which is neither discrete nor Haar measure, and which is invariant under both $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$?