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Dynamics of quadratic polynomials: complex bounds for real maps. (English) Zbl 0881.58053

Summary: 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 \(z\mapsto z^2+c\), \(c\in [-2,1/4]\), is locally connected.

MSC:

37F99 Dynamical systems over complex numbers
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37E99 Low-dimensional dynamical systems
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