  Consider the moduli space, $\mathcal{M}_{d}$, of degree $d \geq 2$
polynomials over $\mathbb{C}$, with a marked critical point. Given $k \geq 0,\;
p$ an odd prime, we show that the set $\Sigma_{k,1,p}$ of conjugacy classes of
degree $p$ polynomials, for which the marked critical point is strictly
$(k,1)$-preperiodic, is an irreducible quasi-affine algebraic set.
Irreducibility of these sets was conjectured by Milnor, and has been proved for
$p=3$ by Buff, Epstein and Koch.
  We prove that the subspaces of $\Sigma_{k,1,p}$, that arise by varying the
ramification index of the marked critical point all the way up to the
unicritical case, are all irreducible subvarieties. Finally, using the
irreducibility of $\Sigma_{k,1,p}$ we give a new and short proof of the fact
that the set of all unicritical points of $\Sigma_{k,1,p}$ form one Galois
orbit under the action of absolute Galois group of $\mathbb{Q}$.

Dynamical Systems