Backward orbits of critical points

We examine the Galois groups of the extensions $K((f'\circ f^n)^{-1}(0))/K$ where $K$ is a number field for polynomials $f(x)\in K[x]$. We use our understanding of this group to study the proportion of primes for which $f$ has a $\mathfrak p$-adic attracting periodic point for a "typical" $f$ and apply the result to the split case of the Dynamical Mordell-Lang Conjecture.