Hausdorff dimension of the limit sets of Tree Iterated Function Systems

Abstract

We investigate Tree Iterated Function Systems (TIFSs), which we introduce in this paper. TIFSs are the generalizations of Iterated Function Systems in which we take the maps independently at each step and each block. In this paper, we give the definition of TIFSs and the limit sets of them. We show a formula for the Hausdorff dimension of the limit sets of TIFSs, which is a generalization of Bowen's formula. Moreover, we give an example which emphasizes the difference between TIFSs and non-autonomous IFSs.