# The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for endomorphisms of (P^1)^n

## TL;DR

This paper proves two major conjectures in arithmetic dynamics for endomorphisms of (P^1)^n, utilizing equidistribution theorems and Julia set symmetry analysis to establish the results.

## Contribution

It establishes the Dynamical Manin-Mumford and Bogomolov Conjectures for a broad class of endomorphisms of (P^1)^n, advancing understanding in arithmetic dynamics.

## Key findings

- Proves the Dynamical Manin-Mumford Conjecture for (P^1)^n
- Proves the Dynamical Bogomolov Conjecture for (P^1)^n
- Uses equidistribution of small height points and Julia set symmetries

## Abstract

We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.