A Refined Conjecture for Factorizations of Iterates of Quadratic Polynomials over Finite Fields

Abstract

Jones and Boston conjectured that the factorization process for iterates of irreducible quadratic polynomials over finite fields is approximated by a one-step Markov model. In this paper, we find unexpected and intricate behavior for some quadratic polynomials, in particular for those whose critical orbits have large cycle and small tail. We also propose a multistep Markov model that explains these new observations better than the model of Jones and Boston.