Arboreal Galois groups of rational maps with nonreal Julia sets.

The article focuses on arboreal Galois groups associated with rational maps whose Julia sets are not contained in the real projective line, particularly at real archimedean places of number fields. It establishes that if a rational map defined over a number field has a nonreal Julia set at such a place, then the corresponding infinite Galois extension generated by backward orbits is nonabelian. The work further characterizes real polynomials whose Julia sets lie entirely in the real line by relating this property to the location of fixed points and the critical interval of the polynomial. In particular, it provides explicit criteria for polynomials of various degrees and leading coefficient signs, including a detailed description for real cubic polynomials with positive leading coefficient. The results contribute to understanding when arboreal Galois representations have abelian image, addressing conjectures in arithmetic dynamics.