INDUCED EXPANSION FOR QUADRATIC POLYNOMIALS

We prove that non-hyperbolic non-renormalizable quadratic polynomials are expansion inducing. For renormalizable polynomials a counterpart o f this statement is that in the case of unbounded combinatorics renorm alized mappings become almost quadratic. The reason for both results i s in the properties of ''box mappings''. This class of dynamical syste ms is systematically studied and the decay of the box geometry is the reason for both results. Specific estimates of the rate of this decay are shown which are sharp in a class of S-unimodal mappings combinator ially related to rotations of bounded type. For real box mappings we u se known methods based on cross-ratios and Schwarzian derivative. To s tudy holomorphic box mapping we introduce a new type of estimates in t erms of moduli of certain annuli which control the box geometry.