arXiv subjects
Publications and source records attributed to Tomasz Dubejko.
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.
A Bl-packing is a (branched) circle packing that ``properly covers'' the unit disc. We establish some fundamental properties of such packings. We give necessary and sufficient conditions for their existence, prove their uniqueness, and show that their underlying surfaces, known as carriers, are quasiconformally equivalent to surfaces of classical Blaschke products. We also extend our earlier approximation results of to general combinatorial patterns of tangencies in Bl-packings. Finally, a branched version of the Discrete Uniformization Theorem of Beardon and Stephenson is given.
It has been shown that univalent circle packings filling in the complex plane $\bold C$ are unique up to similarities of $\bold C$. Here we prove that bounded degree branched circle packings properly covering $\bold C$ are uniquely determined, up to similarities of $\bold C$, 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.
We use recent advances in circle packing theory to develop a constructive method for the approximation of an analytic function $F:Ω\to \bold C$ by circle packing maps providing we have only been given $Ω$, $|F'|\big|_Ω$, and the set of critical points of $F$. This extends the earlier result of Ithiel Carter and Burt Rodin [CR] for $F$ with no critical points.