Simultaneous Rational Periodic Points of Degree-2 Rational Maps

TL;DR

This paper characterizes rational points that are simultaneously periodic under pairs of quadratic maps, assuming conjectures, and establishes bounds on the number of maps sharing such points, with implications for rational dynamics.

Contribution

It provides a complete classification of common rational periodic points for pairs of quadratic maps under certain conjectures, and bounds the number of maps sharing a common periodic point.

Findings

No more than three quadratic maps share a common rational periodic point.

Complete description of pairs of maps with common rational periodic points under conjectures.

Characterization of when a rational point is periodic for infinitely many maps of a specific form.

Abstract

Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in $S$, we give a complete description of triples $(f_1,f_2,p)$ such that $p$ is a rational periodic point for both $f_i\in S$, $i=1,2$. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two nonzero rational numbers $a, b $ are periodic points of the map $\phi_{t_1,t_2}(z)=t_1 z + t_2/z$ for infinitely many nonzero rational pairs $(t_1, t_2)$ if and only if $a^2 = b^2$.