The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps

Abstract

We prove the Dynamical Bogomolov Conjecture for endomorphisms $\Phi :{\mathbb{P}}^1\times {\mathbb{P}}^1\to {\mathbb{P}}^1\times {\mathbb{P}}^1$, where $\Phi (x,y):=(f(x),g(y))$ for any rational functions $f$ and $g$ defined over $\bar{\mathbb{Q}}$. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we can prove the Dynamical Manin–Mumford Conjecture for endomorhisms $\Phi =(f,g)$ of ${\mathbb{P}}^1\times {\mathbb{P}}^1$, where $f$ and $g$ are rational functions defined over an arbitrary field of characteristic 0.