10 problems

Let $\bm M^d$ denote the set of parameters $c$ such that the Julia set of $z\mapsto z^d+c$ is connected; in particular, $\bm M^2$ is the Mandelbrot set. Douady–Hubbard local connec…

Local connectivity conjecture. The boundary $\partial \mathcal H_d$ is locally connected at most maps $f\in \partial \mathcal H_d- \overline{\Phi}(\mathcal R)$. This extends the pr…

Milnor's non-local-connectivity conjecture. If the maps in $\mathcal H$ have an attracting cycle with two distinct free critical points in its attracting basin, then…

Consider the family of unicritical polynomials of degree $d\geq2$ or the exponential family. A parameter fiber is trivial when it contains only the parameter itself; fibers are def…

Let $f$ be a rational map whose Julia set is connected. A periodic critical Fatou component is a periodic Fatou component containing a critical point, and a non-locally-connected p…

Let $f$ be a rational map whose Julia set is connected. A periodic Fatou component is a Fatou component eventually returning to itself under iteration, and a boundary is locally co…

Non-local-connectivity conjecture.

Bromberg's Bers-slice conjecture. The closure of any Bers slice in $QF(S)$ is not locally connected.

Bromberg's surface-group conjecture.

Bromberg's local-connectivity conjecture. Failure of local connectivity is a fairly widespread phenomenon in deformation spaces of hyperbolic $3$-manifolds.