In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with { \it essentially bounded} combinatorics admitting quadratic-like complex extensions. As an application we construct a natural analogue of the period-doubling fixed point. Dynamical hairiness is also proven for maps in this class. These results are proven by analyzing { \it parabolic towers}: sequences of maps related either by renormalization or by { \it parabolic renormalization}.
Categories: math.DS