Settled polynomials over finite fields

We study the factorization into irreducibles of iterates of a quadratic polynomial $f$ over a finite field. We call $f$ settled when the factorization of its $n$th iterate for large $n$ is dominated by ``stable'' polynomials, namely those that are irreducible under post-composition by any iterate of $f$. We prove that stable polynomials may be detected by their action on the critical orbit of $f$ and that the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of $f$. We then define a Markov process based on the critical orbit of $f$ and conjecture that its limiting distribution describes the full factorization of large iterates of $f$. This conjecture implies that almost all quadratic $f$ defined over a finite field are settled. We give several types of evidence for our conjecture.