12 problems

Let $G={\rm SL}(n,\mathbb{R})$, $\Gamma={\rm SL}(n,\mathbb{Z})$, and $X=G/\Gamma$ for $n\geq 3$. Let $A<G$ be the group of positive diagonal matrices with determinant one. A Borel…

Let $n \geq 3$, let $G \cong \operatorname{PGL}_n(\mathbb{R})$, and let $\Gamma < G$ be a lattice as in examples (L-1) or (L-2) of the source. Let $H$ be the relevant diagonal subg…

Measure uniqueness and measurable rigidity conjecture. The set $\mathcal M$ consists of a single measure, and the semiconjugacy $h$ is a measurable isomorphism between the actions…

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for $\mu$-almo…

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for every…

Furstenberg's weak-star convergence conjecture. If $\mu$ is a continuous Borel probability measure on ${\mathbb T}$ that is $T_p$-invariant, then $T_q^n\mu$ converges in the weak-s…

Let $X$ be a $K3$ surface, let $T:X\to X$ be an automorphism with positive topological entropy, let $\mathrm{dVol}=\Omega\wedge\overline{\Omega}$ be the Lebesgue measure, and let…

Let $S,T\colon[0,1)\to[0,1)$ be the maps $Sx=px\pmod 1$ and $Tx=qx\pmod 1$, where $p,q\geq 1$ are multiplicatively independent integers. Let $\mu$ be a Borel probability measure on…

Recall that a strongly simple action has one-dimensional coarse Lyapunov distributions, which integrate to Lyapunov foliations, and that some Lyapunov foliations have conditional m…

Local measure-rigidity conjecture. The action $\alpha'$ has an ergodic invariant measure $\mu'$ satisfying the same assumptions; $\mu'$ can be chosen so that $\mu'\to\mu$ in the we…

Jacobian-rigidity conjecture. Jacobians along Lyapunov foliations for $\alpha$ are rigid.

Let $n\geq 3$, let $X_n$ be the space of lattices of rank $n$, and let $H$ be a maximal $\mathbb{R}$-split torus in $\mathrm{PGL}_n(\mathbb{R})$. Let $\mu$ be an ergodic $H$-invari…