Abstract. Let ϕ : P 1 −→ P 1 be a rational map of degree greater than one defined over a number field k. For each prime p of good reduction for ϕ, we let ϕp denote the reduction of ϕ modulo p. A random map heuristic suggests that for large p, the proportion of periodic points of ϕp in P 1 (o k /p) should be small. We show that this is indeed the case for many rational functions ϕ.

We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not conjugate to a power map, Chebyshev polynomial, or Lattès map. The associated Galois action on an infinite ternary rooted tree has Hausdorff dimension bounded strictly between that of the infinite wreath product of cyclic groups and that of the infinite wreath product of symmetric groups. We deduce a zero-density result for prime divisors in an orbit under this polynomial. We also obtain a zero-density result for the set of places of convergence of Newton's method for a certain cubic polynomial, thus resolving the first nontrivial case of a conjecture of Faber and Voloch.

In 1985, Odoni showed that in characteristic 0 the Galois group of the n-th iterate of the generic polynomial with degree d is as large as possible. That is, he showed that this Galois group is the n-th wreath power of the symmetric group S d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.Several of the results proven in this paper were stated and proven by R.W.K. Odoni in an unpublished preprint, including the polynomial versions of the Galois theoretic results and the application presented in Section 6.2. Although the results and arguments given by Odoni in that manuscript are presented somewhat differently here, his work on this project was invaluable in the completion of this paper.

Given a finite endomorphism ϕ of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K(ϕ −∞ (α)) := n≥1 K(ϕ −n (α)) generated by the preimages of α under all iterates of ϕ. In particular when ϕ is post-critically finite, i.e. there exists a non-empty, Zariski-open W ⊆ X such that ϕ −1 (W ) ⊆ W and ϕ : W → X isétale, we prove that K(ϕ −∞ (α)) is ramified over only finitely many primes of K. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire [AHM05] in the case X = A 1 and Cullinan-Hajir, Jones-Manes [CH12, JM14] in the case X = P 1 . Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P 1 . The proof relies on Faltings' theorem and a local argument.

Let K be a number field, and let d 2.