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

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Kozlovski, O. and van Strien, Sebastian. (2009) Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. Proceedings of the London Mathematical Society, Vol.99 (Part 2). pp. 275-296. ISSN 0024-6115

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Official URL: http://dx.doi.org/10.1112/plms/pdn055

Abstract

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).

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Journal or Publication Title:

Proceedings of the London Mathematical Society

Publisher:

Cambridge University Press

ISSN:

0024-6115