Rational maps with a preperiodic critical point

Abstract

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\ge 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a closely related question concerning the irreducibility (over $\mathbb{Q}$) of the set of conjugacy classes of unicritical polynomials, of degree $D\ge 2$, with a preperiodic critical point. Our proofs are purely algebraic.

Previous version of the paper subsumed by this one