DYNAMICS OF QUADRATIC POLYNOMIALS,

I: COMBINATORICS AND GEOMETRY OF

THE YOCCOZ PUZZLE MIKHAIL LYUBICH

Abstract. This work studies combinatorics and geometry of the Yoccoz puzzle for quadratic polynomials. It is proven that the moduli of the “principal nest” of annuli grow at linear rate. As a corollary we obtain complex a priori bounds and local connectivity of the Julia set for many infinitely renormalizable quadratics.

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