Abstract: TBA Arithmetic dynamics is a field at the intersection of complex dynamics and algebraic number theory. The main questions in arithmetic dynamics are motivated by analogous classical problems in arithmetic geometry, especially the theory of elliptic curves. We study one such question, which is an analogue of Serre's open image theorem regarding $\ell$-adic Galois representations arising from elliptic curves. We consider the action of the absolute Galois group of a field on pre-images of a point $\alpha$ under iterates of a rational map $f$ (points that eventually map to $\alpha$ as we apply $f$ repeatedly). These points can be given the structure of a rooted tree in a natural way. This determines a homomorphism from the absolute Galois group of the field to the automorphism group of this tree, called an arboreal Galois representation.
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