Arboreal Galois representations arise naturally in arithmetic dynamics, when studying iteration of polynomial (or more general) maps. They have recently received a lot of interest, although many of their properties, such as certain potential analogies with elliptic curve Galois representations, remain mysterious.

We explore the implications of new results on "largeness" (in a concrete sense) of arboreal representations on several kinds of problems in arithmetic dynamics. In particular, we discuss applications to problems related to the shape of mod-$p$ forward and backward orbits under iteration of an integer polynomial. Such problems include the problem of "dynamical stability" and the density of prime divisors of dynamical sequences ${f^{circ n}(a): n in mathbb{N}}$.

Based on joint work with Danny Neftin and Shai Rosenberg.