Preperiodic points for families of rational map

TL;DR

This paper investigates conditions under which pairs of points are simultaneously preperiodic for families of rational maps, establishing finiteness results based on polynomial degrees and family parameters.

Contribution

It provides new finiteness theorems for preperiodic points in algebraic families of rational maps, extending previous results to specific polynomial and endomorphism families.

Findings

Finiteness of parameters where both points are preperiodic under certain conditions.

A degree condition on polynomials ensures at most finitely many such parameters.

Extension of results to two-dimensional families of endomorphisms of P^2.

Abstract

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar result for certain two-dimensional families of endomorphisms of P^2.