4 problems

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

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…

Let $f4c2$ be the Mandelbrot set, and let $cf4c2f4c2$. Consider the normalized Lebesgue measure on the disk $f4c2(c,r)$ and the corresponding random iterations of maps of the form…

Let $\mathcal{P}\subseteq\mathbb{R}$ be a Poisson point process. Assign independently to each point $x\in\mathcal{P}$ a random number $N_x\in\{1,2,3,\dots\}$ of marks, and let a pa…