Moduli spaces for families of rational maps on P1

Let ϕ:P1→P1 be a rational map defined over a field K. We construct the moduli space Md(N) parameterizing conjugacy classes of degree-d maps with a point of formal period N and present an algebraic proof that M2(N) is geometrically irreducible for N>1. Restricting ourselves to maps ϕ of arbitrary degree d⩾2 such that h−1○ϕ○h=ϕ for some nontrivial , we show that the moduli space parameterizing these maps with a point of formal period N is geometrically reducible for infinitely many N.