## Institute for Mathematical Sciences

## Preprint ims91-19a

** Y. Jiang**
* On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: I. Maps with Feigenbaum's Topology.*

Abstract: A semigroup (dynamical system) generated by $C^{1+\alpha}$-contracting mappings is considered. We call a
such semigroup regular if the maximum $K$ of the conformal
dilatations of generators, the maximum $l$ of the norms of the
derivatives of generators and the smoothness $\alpha$ of the
generators satisfy a compatibility condition $K< 1/l^{\alpha}$.
We prove the {\em geometric distortion lemma} for a regular
semigroup generated by $C^{1+\alpha}$-contracting mappings.

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