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Recurrent random walks, Liouville's theorem and circle packings

Published online by Cambridge University Press:  01 May 1997

TOMASZ DUBEJKO
TOMASZ DUBEJKO
Affiliation:
Current address: Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A. E-mail address: [email protected]. Mathematical Sciences Research Institute, Berkeley, CA 94720, U.S.A.
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 121 , Issue 3 , May 1997 , pp. 531 - 546
Copyright
Cambridge Philosophical Society 1997

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Footnotes

Research at MSRI is supported in part by grant no. DMS-9022140.