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After giving an introduction to the procedure dubbed slow polynomial mating and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree rational map. We...
It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points. Finally,...
In this paper we study branched coverings of metrized, simplicial trees which arise from polynomial maps with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space compactifying the moduli space of polynomials of degree ; that records the asymptotic behavior of the multipliers of ; and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial...
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