On the quasisymmetrical classification of infinitely renormalizable maps: II. remarks on maps with a bounded type topology.

On the quasisymmetrical classification of infinitely renormalizable maps: I. Maps with Feigenbaum's topology.

A semigroup (dynamical system) generated by $C^{1+α}$-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 $α$ of the generators satisfy a compatibility condition $K< 1/l^α$. We prove the {\em geometric distortion lemma} for a regular semigroup generated by $C^{1+α}$-contracting mappings.