Preperiodic points for families of polynomials

Abstract

Let $a(\lambda )$ and $b(\lambda )$ be two polynomials with coefficients in complex numbers and let $f_{\lamb$ be a one-parameter family of polynomials indexed by all complex numbers $\lambda$. We study whether there exist infinitely many complex numbers $\lambda$ such that both $a(\lambda )$ and $b(\lambda )$ are preperiodic for $f_{\lambda }$.