11 problems

Benedetto–Ih conjecture. If $f_{d,\alpha}$ is not post-critically finite, then there are finitely many parameters $c\in\overline{K}$ that are $S$-integral relative to $(\alpha)$ an…

Let $f \colon {\mathbb{CP}}^n \to {\mathbb{CP}}^n$, with $n \ge 2$, be a post-critically finite endomorphism of degree $d \ge 2$, and let $\lambda$ be an eigenvalue of $f$ along a…

Let $f$ be a post-critically algebraic endomorphism of ${\mathbb{CP}}^n$ with $n\geq 2$ and degree $d\geq 2$. Let $\lambda$ be an eigenvalue of $f$ along a periodic cycle. Hyperbol…

Let $f: \mathbb{C}^n \to \mathbb{C}^n$ be post-critically finite. An affine Weyl iterated-monodromy characterization conjecture asserts that if the iterated monodromy group of $f$…

Let $f$ be a post-critically finite (PCF) holomorphic endomorphism of $\mathbb{P}^k$ of degree at least $2$, and let $J_m$ denote the Julia set of order $m$. Higher-dimensional Jul…

Let $f$ be a post-critically finite (PCF) holomorphic endomorphism of $\mathbb{P}^2$ of degree at least $2$. A sporadic super-saddle fixed point is a fixed point of this type that…

Sparsity conjecture. Let $d\ge3$ and $n\ge2$. For every fixed tail-length $\ell\ge1$, the set

Critical height zero conjecture. The PCF locus in $\operatorname{Pow}^{N}_{d}(\mathbb{C})$ is precisely defined by the equality

Relative rigidity conjecture. Every such family of post-critically finite maps is constant.

Let $\mathcal{M}_2$ be the moduli space of rational maps of degree two. For an integer $n$ and a multiplier $\lambda$, let $\mathrm{Per}_n(\lambda)\subseteq\mathcal{M}_2$ be the cu…

Let $f$ be the post-critically finite endomorphism of $\mathbb{P}\mathbb{C}^2$ considered above, with Julia set $J$. An orbispace metric is a metric on the orbifold-like space obta…