Arboreal Galois groups of rational maps with nonreal Julia sets

TL;DR

This paper investigates the structure of arboreal Galois groups associated with certain rational maps, especially those with non-real Julia sets, revealing non-abelian groups in specific cases including Lattès maps.

Contribution

It establishes non-abelian arboreal Galois groups for certain rational maps with non-real Julia sets and characterizes real Julia sets for polynomials over r.

Findings

Non-abelian arboreal Galois groups for specific maps

Characterization of real Julia sets for polynomials over r

Lattès maps with non-abelian arboreal Galois groups

Abstract

We prove a non-abelian arboreal Galois group result for certain maps with non-real Julia set at an archimedean place. We investigate the question of determining which polynomials defined over $\mathbb{R}$ have real Julia set. Finally we show that for some certain classes of Latt\`es maps associated to the duplication map on an elliptic curve has non-abelian arboreal Galois groups.