On quadratic periodic points of quadratic polynomials

TL;DR

This paper develops a framework to relate periodic points of quadratic polynomials over Galois extensions to those over rationals and provides evidence against the existence of certain periodic points in quadratic fields.

Contribution

It introduces a method to reduce the problem of finding periodic points over Galois extensions to the rational case and investigates the non-existence of period 5 points in quadratic fields.

Findings

No quadratic polynomials (up to conjugation) have period 5 points in quadratic fields.

A new approach links periodic points over extensions to rational points.

Evidence suggests the absence of certain periodic points in quadratic fields.

Abstract

Bounding the number of preperiodic points of quadratic polynomials with rational coefficients is one case of the Uniform Boundedness Conjecture in arithmetic dynamics. Here, we provide a general framework that may reduce finding periodic points of such polynomials over Galois extensions of $\mathbb{Q}$ to finding periodic points over the rationals. Furthermore, we present evidence that there are no such polynomials (up to linear conjugation) with periodic points of exact period 5 in quadratic fields by searching for points on an algebraic curve that classifies quadratic periodic points of exact period 5 and suggesting the application of the method of Chabauty and Coleman for further progress.