The geometry of preperiodic points in families of maps on $\mathbb{P}^N$
Laura DeMarco, Niki Myrto Mavraki
TL;DR
This work develops a unifying conjectural framework to characterize when subvarieties of parameterized projective spaces carry dense sets of preperiodic points under algebraic families of maps. It introduces the relative special dimension $r_{\Φ,\mathcal{X}}$ and the Green current $\hat{T}_{\Φ}$, proving that the density of preperiodic points in $\mathcal{X}$ is equivalent to $\hat{T}_{\Φ}^{\wedge r_{\Φ,\mathcal{X}}}\wedge [\mathcal{X}] \neq 0$, thereby generalizing the Dynamical Manin–Mumford conjecture and connecting to Gao–Habegger’s results on abelian varieties. The paper verifies several key special cases (DMM, $N=1$, DAO) and derives consequences such as sparsity phenomena for special subvarieties and implications for PCF maps in moduli spaces, including a partial proof that links density of preperiodic points to dynamical stability notions. It also outlines how the conjecture implies uniform bounds on shared preperiodic points and frames a path toward a broad dynamical Bogomolov-type theory. Overall, the work offers a cohesive blueprint to understand unlikely intersections in families of dynamical systems on projective spaces with deep ties to height theory and complex dynamics.
Abstract
We study the dynamics of algebraic families of maps on $\mathbb{P}^N$, over the field $\mathbb{C}$ of complex numbers, and the geometry of their preperiodic points. The goal of this note is to formulate a conjectural characterization of the subvarieties of $S \times\mathbb{P}^N$ containing a Zariski-dense set of preperiodic points, where the parameter space $S$ is a quasiprojective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the Relative Manin-Mumford Conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin-Mumford Conjecture (theorem of Raynaud) and the Dynamical Manin-Mumford Conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that several recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.