On the dynamical Bogomolov conjecture for families of split rational maps

We prove that Zhang’s dynamical Bogomolov conjecture holds uniformly along 1-parameter families of rational split maps and curves. This provides dynamical analogues of recent results of Dimitrov, Gao, and Habegger as well as Kühne. In fact, we prove a stronger Bogomolov-type result valid for families of split maps in the spirit of the relative Bogomolov conjecture. We thus provide first instances of a generalization of Baker-DeMarco’s conjecture to higher dimensions. Our proof contains both arithmetic and analytic ingredients. Our main analytic result may be viewed as a dynamical Ax–Lindemann-type theorem for split rational endomorphisms. More precisely, we show that weakly special curves under the action of a split map (f,g) of (PC1)2 are exactly those that lead to linear relations between the measures of maximal entropy of f and g. This extends a previous result of Levin and Przytycki. We further establish a height inequality for families of split maps and varieties comparing the values of a fiber-wise Call–Silverman canonical height with a height on the base and valid for most points of a non-preperiodic variety. This provides a dynamical generalization of a theorem of Haebegger and generalizes results of Call and Silverman as well as Baker to higher dimensions. In particular, we establish a geometric Bogomolov theorem for split rational maps and varieties of arbitrary dimension.