## Institute for Mathematical Sciences

## Preprint ims93-8

** J. Graczyk & G. Swiatek**
* Induced Expansion for Quadratic Polynomials.*

Abstract: 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.

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