Common preperiodic points for quadratic polynomials

TL;DR

This paper establishes a uniform bound on the number of common preperiodic points for different quadratic polynomials, combining arithmetic and complex analysis techniques to achieve effective results with explicit constants.

Contribution

It introduces a novel uniform bound on shared preperiodic points for quadratic polynomials with different parameters, integrating arithmetic and complex-analytic methods.

Findings

Existence of a uniform bound on common preperiodic points

Effective bounds with explicit constants provided

Application of adelic energy pairing and distortion theorems

Abstract

Let $f_c(z) = z^2+c$ for $c \in \mathbb{C}$. We show there exists a uniform bound on the number of points in $\mathbb{P}^1(\mathbb{C})$ that can be preperiodic for both $f_{c_1}$ and $f_{c_2}$ with $c_1\not= c_2$ in $\mathbb{C}$. The proof combines arithmetic ingredients with complex-analytic; we estimate an adelic energy pairing when the parameters lie in $\bar{\mathbb{Q}}$, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proof is effective, and we provide explicit constants for each of the results.