Let be distinct points in , and let be distinct points in . Is there a rational function such that for each ? Thinking of as a monoid under composition, this question asks whether acts -transitively on .
More importantly, if this is false, then is there a nice description of the set of pairs for which there is a rational function such that ?
A rational function is determined by its preimages counting multiplicities at 3 distinct points, so I suspect this is false.