Eventually stable rational functions

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