Settled polynomials over finite fields

Abstract

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