Let a₀ be an integer, let f be a polynomial, and consider the sequence aₙ = f(aₙ₋₁). It is a natural question to ask whether there are infinitely many primes in this sequence. One quickly decides that there are not enough tools at hand to deal with this question and one asks instead about the primes dividing at least one term of the sequence. It turns out that the symmetries of the pre-images of a₀ under the iterates of f play an essential role in the solution of this density question. Motivated by the prime density questions, we study the Galois theory of the iterates of a polynomial (or a rational function). We see fruitful connections between geometry, dynamics, arithmetic, and group theory in this subject.

Some of the results presented in this talk are joint work with Y. Kara, E. Özman.