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Polynomial-like property for real quadratic polynomials. (English) Zbl 1019.37021

Summary: “We prove that all renormalizations of infinitely renormalizable real quadratic polynomials are polynomial-like with the modulus bounded from below by a positive constant, independent of a polynomial. By Douady and Hubbard’s straightening theorem, this means that the renormalizations on appropriately chosen neighborhood of small Julia sets are quasiconformally equivalent to real polynomials and the maximal dilatation of the straightening map is universally bounded.”
This result provides an important part of the proof of the dense hyperbolicity conjecture given in their papers [Ann. Math. (2) 146, 1-52 (1997; Zbl 0936.37015)] and [The real Fatou conjecture, Ann. Math. Stud. 144, Princeton Univ. Press, Princeton (1998; Zbl 0910.30001)]. The crucial part is the almost parabolic case where the folding branch of the induced map does not cover the critical point.
Independent proofs of the universally bounded modulus theorem were given by G. Levin and S. van Strien [Ann. Math. (2) 147, 471-541 (1998; Zbl 0941.37031)] and M. Lyubich and M. Yampolsky [Ann. Inst. Fourier 47, 1219-1255 (1997; Zbl 0881.58053)].

MSC:

37E20 Universality and renormalization of dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37E05 Dynamical systems involving maps of the interval
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37F25 Renormalization of holomorphic dynamical systems
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