Question

Let $f(z)\in\mathbb{C}(z)$ be a rational function, and let $\bar{f}(z)$ denote the function obtained by taking the complex conjugate of the coefficients of $f$. I am interested in maps $f$ for which there is a linear fractional transformation $L(z)=(az+b)/(cz+d)\in\mathbb{C}(z)$ such that $\bar{f}(z)=L\circ f\circ L^{-1}$, but such that there are no $L$ for which $L\circ f\circ L^{-1}$ is in $\mathbb{R}(z)$. It is known that $f$ cannot be a polynomial, and that such maps exist in degree $d$ if and only if $d$ is odd. (These are very special cases of an old result of mine, Compositio Math 98 (1995), 269-304, but they may well have been known before that in this $\mathbb{C}/\mathbb{R}$ setting.) In algebro-geometric terminology, the field of moduli of the map $f$ is not a field of definition. I would like to know if anyone has studied the dynamics of such maps, since dynamically they behave somewhat as if they are defined over $\mathbb{R}$. In particular, do these maps have a name in the dynamics literature? (I did some searching on MathSciNet using terms such as "pseudoreal" and "quasireal", with and without hyphens, but had no success.)