MSc Thesis Defense: Hasan Bilgili, RATIONAL PREPERIODIC POINTS OF QUADRATIC RATIONAL MAPS OVER Q WITH NONABELIAN AUTOMORPHISM GROUPS , Date & Time: 25 June, 2026 – 2:00 PM, Place: FENS G035

RATIONAL PREPERIODIC POINTS OF QUADRATIC RATIONAL MAPS OVER Q WITH NONABELIAN AUTOMORPHISM GROUPS  

Hasan Bilgili
Mathematics, MSc Thesis Dissertation, 2026

Thesis Jury

     Assoc. Prof. Mohammad Sadek (Thesis Advisor)

   Prof. Turgay Bayraktar

  Asst. Prof. Ekin Özman

Date & Time: June 25th, 2026 – 2.00 PM

Place: FENS G035

Keywords : Rational maps of degree $2$, nonabelian automorphism, preperiodic points.

Abstract

In this thesis, we study quadratic rational maps $f:\mathbb{P}^1\to\mathbb{P}^1$ over the rational numbers $\mathbb{Q}$ that have a nonabelian automorphism group. We prove that these maps cannot have a $\mathbb{Q}$-rational periodic point with a period of 4 or more. For maps that do have $\mathbb{Q}$-rational periodic points of period 1, 2, or 3, we provide explicit formulas to describe them. Using these results, we show that any map in this family has a maximum of 6 $\mathbb{Q}$-rational preperiodic points. Finally, we completely classify the portraits of these $\mathbb{Q}$-rational preperiodic points, proving there are exactly 5 possible portraits.