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.)
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asked Jul 8, 2011 at 19:51
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$\begingroup$ There is an analogous theory for subgroups of GL(2,C). Given a finitely generated subgroup of GL(2,C), there is a field generated by the traces of the matrices in the group. One may ask when the subgroup is conjugated to have entries in this field? The answer is determined by looking at a quaternion algebra in M_2(C) generated by linear combinations of the matrices over the trace field. If the algebra splits, then it is a matrix algebra over the field. Otherwise, the quaternion algebra is said to be ramified. The two cases may be determined by considering the Hilbert symbol. $\endgroup$– Ian AgolCommented Jul 9, 2011 at 0:55
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1$\begingroup$ @Agol: Thanks, and similarly for GL(n,C) I assume. For rational maps, the obstruction to the field of moduli being a field of definition is given by an element of the cohomology set $H^1({\rm Gal}(\bar K/K),{\rm PGL}_2)$, so can be related to an element in the Brauer group, hence to the spltting of a quaternion algebra. This is all in my article, and in my book The Arithmetic of Dynamical Systems. (Actually, this is assuming $f$ has no self-similarities, else it's more complicated.) But note that my question is not about the theory, it's asking for references in the (dynamics) literature. $\endgroup$– Joe SilvermanCommented Jul 9, 2011 at 2:48
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