Arboreal Galois groups of rational maps with nonreal Julia sets

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.