## Preprint ims07-04

Jeremy Kahn and Mikhail Lyubich
A priori bounds for some infinitely renormalizable quadratics: III.Molecules.

Abstract: In this paper we prove {\it a priori bounds} for infinitely renormalizable quadraticpolynomials satisfying a molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the satellite types. These {\it a priori bounds} imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.
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