Graph-directed Interated Function Systems Satisfying The Generalized Finite Type Condition

We extend the generalized finite type condition to graph-directed iterated function systems with overlaps. Under this condition, we can compute the Hausdorff dimension of the attractor F in terms of the spectral radius of certain weighted incidence matrix. Moreover, if the Haudorff-dimension of F is a, then the a-dimensional Hausdorff and packing measures of F are shown to be strictly positive. By assuming in addition that the graph is strongly connected, we show that the Hausdorff, packing, and box dimensions are equal and theα-dimensional Hausdorff and packing measures are finite.This thesis consists of five chapters.In chapter one, we recall some notions, backgrounds and facts needed in the sequel, and list the main results of this thesis.In chapter two, we introduce some concepts, such as parent, offspring, neighborhood type, invariant set and so on, also we prove a number of propo-sitions, and finally set up the graph-directed generalized finite type condition.In chapter three, we construct the weighted incidence matrix Aa, intro-duce the measure, and at last prove the main results of this article by using the knowledge of real analysis, fractal geometry and so on.In chapter four, by applying the main results, we solve some interesting problems which cannot be dealt with by methods developed before.In chapter five, two questions are given.