Eventually stable rational functions

Abstract

For a field [Formula: see text], rational function [Formula: see text] of degree at least two, and [Formula: see text], we study the polynomials in [Formula: see text] whose roots are given by the solutions in [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th iterate of [Formula: see text]. When the number of irreducible factors of these polynomials stabilizes as [Formula: see text] grows, the pair [Formula: see text] is called eventually stable over [Formula: see text]. We conjecture that [Formula: see text] is eventually stable over [Formula: see text] when [Formula: see text] is any global field and [Formula: see text] is any point not periodic under [Formula: see text] (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when [Formula: see text] has a discrete valuation for which (1) [Formula: see text] has good reduction and (2) [Formula: see text] acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of [Formula: see text]-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.