Arithmetic properties of $3$-cycles of quadratic maps over $\mathbb{Q}$

TL;DR

This paper investigates the arithmetic properties of rational 3-cycles in quadratic maps over the rationals, identifying unique minimal numerator and denominator cases and exploring the structure of rational periodic points.

Contribution

It establishes the uniqueness of certain rational parameters with minimal numerator and denominator for 3-periodic points and analyzes the arithmetic structure of these points and their orbits.

Findings

$c=-29/16$ is the unique rational with smallest denominator for 3-periodic points

$c=-29/16$ is also the unique rational with smallest numerator for 3-periodic points

A graph structure on numerators reflects solutions to norm equations in a cubic field

Abstract

It is shown that $c=-29/16$ is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map $f_c(x) = x^2+c$ has a rational periodic point of period $3$. Several arithmetic conditions on the set of all such rational numbers $c$ and the rational orbits of $f_c(x)$ are proved. A graph on the numerators of the rational $3$-periodic points of maps $f_c$ is considered which reflects connections between solutions of norm equations from the cubic field of discriminant $-23$.