On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections
On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections
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DOI：
10.1090/ecgd/379
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发表时间：
2019-12
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影响因子：
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通讯作者：
Russell Lodge;M. Lyubich;S. Merenkov;S. Mukherjee
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文献类型：
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作者：
Russell Lodge;M. Lyubich;S. Merenkov;S. Mukherjee
根据圆填充定理，黎曼球的任何三角剖分都可以实现为一个圆填充的神经。对偶圆上的反射产生一个Kleinian群H H，其极限集是广义阿波罗垫片Λ H \Lambda _H。我们设计了一个将H H与一个有理映射g g联系起来的手术，其Julia集合J g \mathcal {J}_g（非拟共形）同胚于Λ H \Lambda _H。然而，我们证明了对于一大类三角剖分，Λ H \Lambda _H和J g \mathcal {J}_g的拟对称群是同构的，并且与相应的自同胚群重合。而且，在H H的情况下，这个群等于Λ H \Lambda _H的Möbius对称群，它是H H本身与底层圆填充的Möbius对称群的半直接积。在四面体三角剖分的情况下（当Λ H \Lambda _H为经典阿波罗衬垫时），我们给出了上述作用的拟正则模型，该模型拟共形等价于g g，并通过David手术产生H H。我们还构造了群与同一动力平面上共存的映射之间的匹配，并证明了它可以由三角线和内切圆上的Schwarz反射产生。
According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H H whose limit set is a generalized Apollonian gasket Λ H \Lambda _H . We design a surgery that relates H H to a rational map g g whose Julia set J g \mathcal {J}_g is (non-quasiconformally) homeomorphic to Λ H \Lambda _H . We show for a large class of triangulations, however, the groups of quasisymmetries of Λ H \Lambda _H and J g \mathcal {J}_g are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H H , this group is equal to the group of Möbius symmetries of Λ H \Lambda _H , which is the semi-direct product of H H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when Λ H \Lambda _H is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to g g and produces H H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.