We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.

it is conjectured that a rational map whose coefficients are algebraic over $\mathbb q_p$ has no wandering components of the fatou set. benedetto has shown that any counterexample to this conjecture must have a wild recurrent critical point. we provide the first examples of rational maps whose coefficients are algebraic over $\mathbb q_p$ and that have a (wild) recurrent critical point. in fact, it is shown that there is such a rational map in every one-parameter family of rational maps that is defined over a finite extension of $\mathbb q_p$ and that has a misiurewicz bifurcation.