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Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

By O. Kozlovski and Sebastian van Strien


We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by-product we prove that the Julia set of a non-renormalizable polynomial with only hyperbolic periodic points is locally connected, and the Branner-Hubbard conjecture. The main tools are the enhanced nest construction (developed in a previous joint paper with [Rigidity for real polynomials, Ann. of Math. (2) 165 (2007) 749-841.]) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case)

Topics: QA
Publisher: Cambridge University Press
Year: 2009
DOI identifier: 10.1112/plms/pdn055
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