Premium
Local connectivity and quasi‐conformal rigidity of non‐renormalizable polynomials
Author(s) -
Kozlovski Oleg,
van Strien Sebastian
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn055
Subject(s) - rigidity (electromagnetism) , conformal map , conformal symmetry , pure mathematics , mathematics , physics , geometry , quantum mechanics
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).