Simultaneous rational periodic points of degree-2 rational maps

Let S be the collection of quadratic polynomial maps, and degree 2-rational maps whose automorphism groups are isomorphic to C2 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 (f1, f2, p) such that p is a rational periodic point for both fi is an element of 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 t1,t2(z) = t1z + t2/z for infinitely many nonzero rational pairs (t1, t2) if and only if a2 = b2.(c) 2022 Elsevier Inc. All rights reserved.