Dynamics of certain non-conformal semigroups

Abstract A semigroup generated by two dimensional C contracting maps 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. We prove that the shape of the image of the core of a ball under any element of a regular semigroup is good (bounded geometric distortion like the Koebe 1/4-lemma [1]). And we use it to show a lower and a upper bounds of the Hausdorff dimension of the limit set of a regular semigroup. We also consider a semigroup generated by higher dimensional maps.