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

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 polynomials 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 Weixiao Shen) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case).