The algebraic dynamics of generic endomorphisms of P^n

TL;DR

This paper proves that generic endomorphisms of projective spaces over characteristic zero fields exhibit properties like no non-trivial preperiodic subvarieties and Zariski dense preperiodic points, confirming key conjectures in algebraic dynamics.

Contribution

It establishes the dynamical Manin–Mumford and Mordell–Lang conjectures for generic endomorphisms of projective spaces over characteristic zero fields.

Findings

Generic endomorphisms have no non-trivial preperiodic subvarieties.

Preperiodic points form a Zariski dense set.

Infinite orbits are Zariski dense.

Abstract

We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no non-trivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski dense and any infinite subset of a single orbit is also Zariski dense, thereby verifying the dynamical "Manin--Mumford" conjecture of Zhang and the dynamical "Mordell--Lang" conjecture of Denis and Ghioca--Tucker in this case.