5 problems

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…

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.

Escaping-set conjecture. If $f\in\mathcal{B}$ is a transcendental entire function, then $I(f)$ is not a spider's web.

Let $f$ be a transcendental entire function, and let $I(f)$ denote its escaping set. A curve in $I(f)$ joins a point to infinity if it is a curve contained in $I(f)$ whose endpoint…

Let $f(z)=e^z$ be the exponential map, and let $J(f)$ denote its Julia set on the Riemann sphere $\widehat{\mathbb C}$. Fatou's conjecture. … This conjecture asserts that the expon…