1 0 M ay 1 99 5 RECURRENT RANDOM WALKS , LIOUVILLE ’ S THEOREM , AND CIRCLE PACKINGS
نویسندگان
چکیده
It has been shown that univalent circle packings filling in 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.
منابع مشابه
1 6 A ug 1 99 5 RANDOM WALKS OF CIRCLE PACKINGS
A notion of random walks for circle packings is introduced. The geometry behind this notion is discussed, together with some applications. In particular, we obtain a short proof of a result regarding the type problem for circle packings, which shows that the type of a circle packing is closely related to the type of its tangency graph.
متن کاملRandom Walks , Liouville ' S Theorem , and Circle Packings
It has been shown that univalent circle packings lling in 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 packin...
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A notion of random walks for circle packings is introduced. The geometry behind this notion is discussed, together with some applications. In particular, we obtain a short proof of a result regarding the type problem for circle packings, which shows that the type of a circle packing is closely related to the type of its tangency graph.
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