On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections

<p>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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose limit set is a generalized Apollonian gasket <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ<! -- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We design a surgery that relates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a rational map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose Julia set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="script upper J Subscript g"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {J}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is (non-quasiconformally) homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ<! -- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show for a large class of triangulations, however, the groups of quasisymmetries of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ<! -- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="script upper J Subscript g"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {J}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this group is equal to the group of Möbius symmetries of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ<! -- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is the semi-direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ<! -- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and produces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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. </p>