Pseudo Real Rational Maps and their automorphisms
A rational maps is called real if it is definable over the reals, equivalently, if it admits a reflection as an anticonformal automorphism. A rational map is called pseudo real if it admits antiholomorphic automorphisms but it is not real. Real and pseudo real maps are exactly those rational maps with the property that the field of moduli (the intersection of all fields of definition) is a subfield of the reals. Pseudo real rational maps are the only known examples of rational maps which cannot be defined over their field of moduli. We observe that the group of holomorphic automorphisms of a pseudo real rational map is either trivial or cyclic and we provide a characterization of pseudo real rational maps in terms of the group of holomorphic automorphisms. The non connectivity in moduli space of the locus of pseudo real rational maps is also obtained.