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Title:
David homeomorphisms and applications in mating and removability
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Abstract:
The main object in this talk will be the mating of piecewise (anti-)analytic dynamical systems of the unit disk. While quasiconformal maps can be used for the mating of two hyperbolic dynamical systems, they are insufficient for mating a hyperbolic dynamical system with a parabolic one. Instead, we achieve the mating using the notion of a David homeomorphism, which is a generalization of a quasiconformal homeomorphism that allows unbounded quasiconformal dilatation. The main theorem that we will discuss provides extensions of a general class of dynamically defined circle homeomorphisms to David homeomorphisms of the unit disk. An implication of this theory is that limit sets of a certain class of Kleinian reflection groups (called necklace reflection groups) are conformally removable. This is joint work with Misha Lyubich, Sergei Merenkov, and Sabyasachi Mukherjee.
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