9 problems

Eremenko's conjecture. Each connected component of $I(f)$ is unbounded.

Consider the cosine family of entire maps … A parameter is Misiurewicz when its singular values have the corresponding Misiurewicz behavior, and two maps are topologically conjugat…

Let $E_{\kappa}(z)=e^z+\kappa$ be an exponential map, let $J(E_{\kappa})$ denote its Julia set, and let $I(E_{\kappa})$ denote its escaping set. An order-preserving conjugacy betwe…

Let $f$ and $g$ be two distinct permutable entire functions, and let $Q$ be a non-constant polynomial. Suppose that … where $a\in\mathbb{C}\setminus\{0\}$ and $b\in\mathbb{C}$. Wan…

Sixsmith's conjecture. The escaping set $I(f)$ is not a spider's web for any $f\in\mathcal{B}$.

For a transcendental entire function $f$, let … be its escaping set. Escaping-set connectedness dichotomy conjecture. In general, $I(f)$ is either connected or has uncountably many…

Let $f:\to$ be a transcendental self-map of the punctured plane, where $=\setminus\{0\}$. A fast escaping spider's-web conjecture. The fast escaping set $A(f)$ is not a … sufficien…

Quite fast escaping set conjecture. The set $Q(f)$ contains a spider's web. Consequently, $I(f)$ is a spider's web and is connected.

Let $p$ be a quadratic polynomial with a repelling fixed point $z_0$, and let $L$ be a linearizer of $p$ at $z_0$, satisfying … Assume that the critical point of $p$ escapes under…