On the quasisymmetrical classification of infinitely renormalizable   maps: II. remarks on maps with a bounded type topology.
  We use the upper and lower potential functions and Bowen's formula estimating the Hausdorff dimension of the limit set of a regular semigroup generated by finitely many $C^{1+\alpha}$-contracting mappings. This result is an application of the geometric distortion lemma in the first paper at this series. 
Yunping Jiang