Rational maps whose complex conjugate equals a PGL conjugate

2011-07-08T19:51:52Z

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.)

Rational maps whose complex conjugate equals a PGL conjugate - MathOverflow

Rational maps whose complex conjugate equals a PGL conjugate