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

### Dragos Ghioca

University of British Columbia, Vancouver, Canada### Khoa D. Nguyen

University of British Columbia and Pacific Institute for the Mathematical Sciences, Vancouver, Canada### Hexi Ye

Zhejiang University, Hangzhou, China

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## 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.

## Cite this article

Dragos Ghioca, Khoa D. Nguyen, Hexi Ye, The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1571–1594

DOI 10.4171/JEMS/869