In classical real and complex dynamics, one studies topological and analytic properties of orbits of points under iteration of self-maps of $\mathbb R$ or $\mathbb C$ (or more generally self-maps of a real or complex manifold). In arithmetic dynamics, a more recent subject, one likewise studies properties of orbits of self-maps, but with a number theoretic flavor. Many of the motivating problems in arithmetic dynamics come via analogy with classical problems in arithmetic geometry: rational and integral points on varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational maps; and abelian varieties with complex multiplication correspond to post-critically finite rational maps.

This analogy focuses on forward iteration, but sometimes surprising and interesting results can be found by thinking instead about pre-images of rational points under iteration. In this talk, we will give some background and motivation for the field of arithmetic dynamics in order to describe some of these "backwards iteration" results, including uniform boundedness for rational pre-images and open image results for Galois representations associated to dynamical systems.