Irreducibility of preperiodic curves in the moduli space of cubic polynomials
Consider the moduli space, $\mathcal{M}_{3},$ of cubic polynomials over $\mathbb{C}$, with a marked critical point. Let $\Sigma_{k,n}$ be the set of all points for which the marked critical point is strictly $(k,n)$-preperiodic. Milnor conjectured that $\Sigma_{k,n}$'s are irreducible curves in $\mathcal{M}_{3}$, for all $k \geq 0, n>0$. Buff, Epstein, and Koch have proved this conjecture for $k\geq 0, n=1$. In this article, we show the irreducibility of $\Sigma_{k,2}, k\geq 0$ curves. The curves, $\Sigma_{k,2}, k\geq 0$, exhibit a splitting-merging phenomenon that does not occur for $\Sigma_{k,1}$ curves. Furthermore, using the irreducibility of $\Sigma_{k,2}$ curves, we prove an irreducibility result in the unicritical cubic case. Stronger versions of this result in the unicritical cubic case, have been proved by Vefa Goksel and Buff, Epstein, and Koch, but our methods are different. Finally, we show that our method does not extend directly for $\Sigma_{k,q}$ curves, where $q$ is an odd prime.