The geometry of preperiodic points in families of maps on $\mathbb{P}^N$

TL;DR

This paper explores the geometric structure of preperiodic points in algebraic families of maps on projective space, proposing a conjecture linking dense preperiodic points to invariant Green currents, extending classical conjectures in number theory and dynamics.

Contribution

It formulates a conjectural characterization of subvarieties with dense preperiodic points in families of maps, generalizing the Manin-Mumford and Dynamical Manin-Mumford conjectures.

Findings

Proposes a conjecture relating dense preperiodic points to Green currents.

Provides examples where the conjecture holds.

Proves one direction of the conjectural characterization.

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.