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Rational maps with a preperiodic critical point
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abstract
We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, 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 $D\geq 2$, with a preperiodic critical point. Our proofs are purely algebraic.
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A note on Misiurewicz polynomials
For prime d, the Misiurewicz polynomial G_{d,m,n} has no more rational irreducible factors than its reduction modulo d, which yields new irreducibility families and the first bounds for periods above three.
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