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.