Abstract
It has been shown that univalent circle packings filling the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result.We also establish a circle packing analogue of Liouville's theorem: if f is a circle packing map whose domain packing is infinite, univalent, and has recurrent tangency graph, then the ratio map associated with f is either unbounded or constant.
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