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

Abstract

We prove the Dynamical Bogomolov Conjecture for endomorphisms $\Phi:\mathbb P^1\times \mathbb P^1\lra \mathbb P^1\times \mathbb P^1$, 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.