The Properties Of Infinite Iterated Systems Of Contracting Similitudes

In this paper we disscuss some separation properties of the infinite iterated systemsof contracting similitudes. A sufficient condition for the iterated systems satisfying thefinite strong open condition is given. A formula determining the Hausdorff dimension ofthe invariant set is obtained. We also disscuss the upper convex density of the invariantset. We show that for arbitrary fixed point x in the invariant set J there exists a compactset V(?)J with |V|＞0 and x∈V such that (?)_c~s(J, x)=H~s(V)/|V|~s, where the iteratedsystem satisfies the finite strong separation condition and J is closed.