On the dynamical Bogomolov conjecture for families of split rational maps

TL;DR

This paper proves a uniform version of Zhang's dynamical Bogomolov conjecture for families of split rational maps, extending previous results and providing new height inequalities and characterizations of preperiodic curves in higher dimensions.

Contribution

It establishes a stronger Bogomolov-type result for families of split maps, generalizing conjectures by Baker and DeMarco to higher dimensions with new arithmetic and analytic techniques.

Findings

Proved a uniform dynamical Bogomolov conjecture for split rational maps.

Characterized preperiodic curves under non-exceptional split rational endomorphisms.

Established a height inequality linking fiber-wise canonical heights with base heights.

Abstract

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-Habegger and K\"uhne. 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 a conjecture by Baker and DeMarco to higher dimensions. Our proof contains both arithmetic and analytic ingredients. We establish a characterization of curves that are preperiodic under the action of a non-exceptional split rational endomorphism $(f,g)$ of $(\mathbb{P}^1_{\mathbb{C}})^2$ with respect to the measures of maximal entropy of $f$ and $g$, extending a previous result of Levin-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 result by Habegger and generalizes results of Call-Silverman and Baker to higher dimensions. In particular, we establish a geometric Bogomolov theorem for split rational maps and varieties of arbitrary dimension.