3 problems

Positive-measure disconnectedness conjecture. There exists a set of positive Lebesgue measure $\Lambda\subset\mathbb T$ such that $\mathcal J_u$ is disconnected for all…

Almost-everywhere zero-measure conjecture. For Lebesgue almost all $u$, $\mathcal J_u$ has zero Lebesgue measure.

Let $\Gamma<\operatorname{PSL}_2(\mathbb{R})$ be a lattice, and let $m$ be a finitely supported measure whose support generates $\Gamma$. Let the corresponding stationary measure b…