Abstract

. We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map z 7! z 2 + c, c 2 [\Gamma2; 1=4], is locally connected. 1. Introduction Complex a priori bounds proved to be a key analytic issue of the Renormalization Theory. They lead to rigidity results, local connectivity of Julia sets and the Mandelbrot set, and convergence of the renormalized maps (see [HJ, L2, McM1, McM2, MS, R, S]). By definition, an infinitely renormalizable map f has complex bounds if all its renormalizations R n f extend to quadratic-like maps with definite moduli of the fundamental annuli. Sullivan established this property for real infinitely renormalizable maps with bounded combinatorics (see [S] and [MS]). On the other hand, it was shown in [L1, L2] that the...

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