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A priori bounds for some infinitely renormalizable quadratics. II: Decorations. (English) Zbl 1156.37311
Summary: A decoration of the Mandelbrot set \(M \) is a part of \(M \) cut off by two external rays landing at some tip of a satellite copy of \(M \) attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we prove a priori bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.
Part I, see J. Kahn, A priori bounds for some infinitely renormalizable quadratics. I: Bounded primitive combinatorics. Preprint Stony Brook, # 5 (2006).

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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