Quadratic rational functions with a rational periodic critical point of period 3

TL;DR

This paper classifies quadratic rational functions over rationals with a rational periodic critical point of period 3, identifying six possible preperiodic point graphs and bounding the number of such points.

Contribution

It provides a complete classification of preperiodic point graphs under specific conditions, assuming no points of period five or more and a conjecture on a genus 6 curve.

Findings

Exactly six possible preperiodic point graphs.

Rational functions under these conditions have at most eleven rational preperiodic points.

The classification relies on two key assumptions.

Abstract

We provide a complete classification of possible graphs of rational preperiodic points of quadratic rational functions defined over the rationals with a rational periodic critical point of period 3, under two assumptions: that these functions have no periodic points of period at least 5 and the conjectured enumeration of points on a certain genus 6 affine plane curve. We show that there are exactly six such possible graphs, and that rational functions satisfying the conditions above have at most eleven rational preperiodic points.