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Complex a priori bounds revisited. (English) Zbl 1070.37029
The author studies the existence of complex a priori bounds for renormalizations of real quadratic polynomials. He introduces the combinatorial condition of essentially bounded type, which was the subject studied by the author and M. Lyubich [Ann. Inst. Fourier (Grenoble) 47, 1219–1255 (1997; Zbl 0881.58053)] and gives a new treatment to polynomials satisfying this condition. The approach used in the paper is to consider them as small perturbations of parabolic maps, and to use the rigidity properties of such maps to pass from real a priori bounds to complex ones.
37E20Universality, renormalization
37F50Small divisors, rotation domains and linearization; Fatou and Julia sets
30D05Functional equations in the complex domain, iteration and composition of analytic functions