6 problems

Let $f$ be a transcendental entire function, and let … be its escaping set. Eremenko's conjecture. Every connected component of $I(f)$ is unbounded. The conjecture has been confirm…

Let $f:\mathbb{C}\to\mathbb{C}$ be the exponential map $f(z)=e^z$, and let its hairs denote the corresponding invariant curves in the escaping set. Real-analyticity conjecture for…

Let $E(e^z)=\{z\in\mathbb{C}: e^{z},e^{e^{z}},\ldots,e^{\circ n}(z)\to\infty\}$ be the escaping set of the map $z\mapsto e^z$. Escaping-set intersection conjecture. Every line segm…

Forward-orbit stable-basin conjecture. For each component $U$ of $F(S)$, there is a stable basin $V$ for $S$ lying in the forward orbit of $U$ that has cofinite index in $S$.

Stable-basin conjecture. Then a stable basin $U$ has cofinite index in $S$.

Escaping endpoints always explode. The set $\tilde{E}(f_a)$ is totally separated, and hence $\infty$ is an explosion point of $\tilde{E}(f_a)\cup\{\infty\}$.