Iterated function system and Galton-Watson tree

Abstract Given a system {S 1 ,…,S N } 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 μ ω supported on random subset K ω of K, μ ω having some “non-standard” multifractal structure, which contrasts the well-known multifractal formalism for the invariant measure of system {S 1 ,…,S N } may possess. The main tool is the multifractal structures of a Galton-Watson tree, which are obtained by Liu [9], Shieh-Taylor [14], and Morters-Shieh [12].