Quadratic maps with a periodic critical point of period 2

TL;DR

This paper classifies the possible structures of rational preperiodic points for quadratic maps with a specific periodic critical point, extending previous results and limiting the number of such points.

Contribution

It provides a complete classification of graphs of rational preperiodic points for quadratic maps with a period-2 critical point, under certain restrictions, extending Poonen's work.

Findings

13 possible preperiodic graph structures

Maps have at most 9 rational preperiodic points

Data on graphs with critical points of periods 3 or 4

Abstract

We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are 13 possible graphs, and that such maps have at most 9 rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.