# Journal of the European Mathematical Society

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**Volume 21, Issue 5, 2019, pp. 1571–1594**

**DOI: 10.4171/JEMS/869**

Published online: 2019-02-01

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

Dragos Ghioca^{[1]}, Khoa D. Nguyen

^{[2]}and Hexi Ye

^{[3]}(1) University of British Columbia, Vancouver, Canada

(2) University of British Columbia and Pacific Institute for the Mathematical Sciences, Vancouver, Canada

(3) Zhejiang University, Hangzhou, China

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.

*Keywords: *Dynamical Manin–Mumford Conjecture, equidistribution of points of small height, symmetries of the Julia set of a rational function

Ghioca Dragos, Nguyen Khoa, Ye Hexi: The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps. *J. Eur. Math. Soc.* 21 (2019), 1571-1594. doi: 10.4171/JEMS/869