circle packings
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2022 ◽  
Vol 10 ◽  
Author(s):  
Russell Lodge ◽  
Yusheng Luo ◽  
Sabyasachi Mukherjee

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.


Author(s):  
David Krieg ◽  
Elias Wegert
Keyword(s):  

2019 ◽  
Vol 50 (8) ◽  
pp. 1061
Author(s):  
Lan Shiyi ◽  
Liu Jinsong

2019 ◽  
Vol 372 (9) ◽  
pp. 6231-6261
Author(s):  
Huabin Ge ◽  
Wenshuai Jiang
Keyword(s):  

2019 ◽  
Vol 155 (6) ◽  
pp. 1118-1170
Author(s):  
Elena Fuchs ◽  
Katherine E. Stange ◽  
Xin Zhang

We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group ${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$ satisfying certain conditions, where $K$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that ${\mathcal{A}}$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $\operatorname{PSL}_{2}({\mathcal{O}}_{K})$ containing a Zariski dense subgroup of $\operatorname{PSL}_{2}(\mathbb{Z})$.


Soft Matter ◽  
2019 ◽  
Vol 15 (15) ◽  
pp. 3076-3084 ◽  
Author(s):  
Varda F. Hagh ◽  
Eric I. Corwin ◽  
Kenneth Stephenson ◽  
M. F. Thorpe

Jammed packings can be generated by pruning elastic networks and mapping them into circle packings.


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