In this thesis, we study the dynamics of holomorphic mappings from the unit disk in the complex plane back to itself. We first look at the properties of automorphisms of the unit disc and then move on to consider the convergence of sequences of holomorphic mappings. Using the generalizations of the Schwarz lemma to the boundary of the unit disk, we prove the Dejoy-Wolff theorem, which tells us that the sequence of iterates of an arbitrary holomorphic mapping from the unit disk, without fixed points in it, converges on compact sets to a point on the boundary of this disk.
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