Abstract
Abstract In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.
Highlights
Analogies between the two branches of conformal dynamics, which are commonly known as Sullivan’s dictionary, provide a conceptual framework for understanding the connections but motivate research in each field as well
◦ Kissing reflection groups: groups generated by reflections along the circles of finite circle packings P and
We show that the planar dual of T is a 2-connected simple plane graph
Summary
A circle packing P is a connected finite collection of (oriented) circles in C with disjoint interiors. It can be checked that the contact graph of a circle packing is simple. This turns out to be the only constraint for the graph (See [44, Chapter 13]). Every connected simple plane graph is isomorphic to the contact graph of some circle packing. An infinite circle packing P is an infinite collection of (oriented) circles in Cwith disjoint interiors, whose contact graph is connected. Suppose Γ is a polyhedral graph; there is a pair of circle packings whose contact graphs are isomorphic to Γ and its planar dual. A circle packing P is said to be marked if the associated contact graph is marked
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