Finite orbit points for sets of quadratic polynomials

TL;DR

This paper classifies rational points with finite orbits under sets of quadratic polynomials with bounded period length, showing that for four or more polynomials, such points do not exist, and proposes an analog of a conjecture in this context.

Contribution

It provides a classification of finite orbit points for sets of quadratic polynomials with bounded periods and formulates a new conjecture analogous to Morton-Silverman for these sets.

Findings

No finite orbit points exist for sets of four or more quadratic polynomials.

Complete classification of pairs (S, P) with finite orbits under the given conditions.

Formulation of an analog of the Morton-Silverman Conjecture for sets of rational functions.

Abstract

Let $S=\{x^2+c_1, x^2+c_2,\dots, x^2+c_s\}$ be a set of quadratic polynomials with rational coefficients, and let $P$ be a rational basepoint. We classify the pairs $(S,P)$ for which $P$ has finite orbit for $S$, assuming that the maximum period length for each individual polynomial is at most three (conjectured by Poonen). In particular, under these hypotheses we prove that if $s\geq4$, then there are no points $P$ with finite orbit for $S$. Moreover, we use this perspective to formulate an analog of the Morton-Silverman Conjecture for sets of rational functions.