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Proceedings of the London Mathematical Society Advance Access first published online on February 26, 2009
This version published online on July 21, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn055
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© 2009 London Mathematical Society

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

Oleg Kozlovski and Sebastian van Strien

Mathematics Institute
University of Warwick
Zeeman Building
Coventry
CV4 7AL
United Kingdom
strien@maths.warwick.ac.uk

Received 14 March 2008. Revision received 21 October 2008.

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).

2000 Mathematics Subject Classification 37F10, 37F15, 37F30.

The second authors of this paper was partially supported by CODY and by a Leverhulme Trust Senior Fellowship.

The first version was incorrect. The definition of combinatorial equivalence was rectified in the current version.


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