  We prove that non-hyperbolic non-renormalizable quadratic polynomials are
expansion inducing. For renormalizable polynomials a counterpart of this
statement is that in the case of unbounded combinatorics renormalized mappings
become almost quadratic. Technically, this follows from the decay of the box
geometry. Specific estimates of the rate of this decay are shown which are
sharp in a class of S-unimodal mappings combinatorially related to rotations of
bounded type. We use real methods based on cross-ratios and Schwarzian
derivative complemented by complex-analytic estimates in terms of conformal
moduli.
