Preperiodic points for families of polynomials

Abstract

Let $a\left(\lambda \right),b\left(\lambda \right)\in ℂ\left[\lambda \right]$, and let $f_{\lambda }\left(x\right)\in ℂ\left[x\right]$ be a one-parameter family of polynomials indexed by all $\lambda \in ℂ$. We study whether there exist infinitely many $\lambda \in ℂ$ such that both $a\left(\lambda \right)$ and $b\left(\lambda \right)$ are preperiodic for $f_{\lambda }$.