Non-Archimedean Rigidity and the Uniform Dynamical Manin--Mumford for $\left(\mathbb{P}^1\right)^n$

Source abstract

We prove a uniform form of the dynamical Manin-Mumford conjecture for split endomorphisms $Φ= \left(f_1,\ldots,f_n\right): \left(\mathbb{P}^1\right)^n \to \left(\mathbb{P}^1\right)^n$ over $\mathbb{C}$, where each $f_i$ has degree $d \geq 2$ and at most one $f_i$ is exceptional. Specifically, we bound the number of irreducible components of $\overline{\operatorname{Prep}\left(Φ\right)\cap V\left(\mathbb{C}\right)}^{\operatorname{Zar}}$ uniformly, depending only on $\left(n,d,\operatorname{deg} V\right)$. We also obtain uniform statements for the dynamical Bogomolov conjecture. Our approach combines a degeneration argument via ultrafilters with a new non-archimedean rigidity theorem, along with the quantitative dynamical Zhang's inequality by Mavraki and the second author.