A subscription is required to access this article.
We prove the Dynamical Bogomolov Conjecture for endomorphisms , where for any rational functions and defined over . 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 of , where and 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. 1571–1594DOI 10.4171/JEMS/869