Iterated function system and Galton-Watson tree

Given a system {S1,…,SN} of N contractive similarities satisfying some strong separation condition, it has an invariant set K for the system. In this article, the authors construct some random measure μω supported on random subset Kω of K, μω having some “non-standard” multifractal structure, which contrasts the well-known multifractal formalism for the invariant measure of system {S1,…,SN} may possess. The main tool is the multifractal structures of a Galton-Watson tree, which are obtained by Liu [9], Shieh-Taylor [14], and Mörters-Shieh [12].