Rational Functions with a Unique Critical Point

Over an algebraically closed field of positive characteristic, there exist rational functions with extreme ramification: only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we use this tool to show that the space of unicritical functions is rational and geometrically irreducible and compute its dimension. We exhibit similar results for quotients of the space of unicritical functions by the SL2-actions of conjugation and postcomposition.