The Geometric Dynamical Northcott Property in the quadratic family

TL;DR

This paper provides a simplified proof of the geometric Northcott property within the quadratic family of dynamical systems, focusing on the one-dimensional case to clarify the underlying ideas.

Contribution

It offers a clearer, more explicit proof of the geometric Northcott property for quadratic polynomials, making the concepts more accessible.

Findings

Simplified proof of Theorem A for quadratic family

Explicit parametrization aids understanding of the geometric Northcott property

Clarifies the dynamics in one-dimensional quadratic systems

Abstract

The aim of this note is to give a proof of Theorem A from our work on the geometric Northcott property in the simpler case of the quadratic family; being in dimension $1$ in both the dynamical space and the parameter space, and having a simple and explicit parametrization of the family allow to simplify the proof and, we hope, make the ideas more apparent.