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 nth 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 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. Let Fq be the finite field with q elements, and consider a polynomial f ∈ Fq[x] of degree d. In this paper we consider the iterates of f , namely the polynomials f := f ◦ f ◦ · · · ◦ f } {{ }