Simultaneous Rational Periodic Points of Degree-2 Rational Maps

2022/06/07 by Burcu Barsakçı, Barsakçı, Burcu, Mohammad Sadek +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2206.03526

openalex publication_date 2022/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 f_i∈ 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 ϕ_t1,t2(z)=t₁ z + t₂/z for infinitely many nonzero rational pairs (t₁, t₂) if and only if a² = b².

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