Digraph Self-similar Set Of Multifractal Analysis

This thesis contains two main topics related to fractal geometry and dynamical systems.The first chapter is about graph-directed self-similar sets in complete metric spaces, and the second chapter is about multifractal analysis of equilibrium measures in dynamical systems.In Chapter One,we find the relationship between strong open set condition, Hausdorff dimension,the positivity of the Hausdorff measure and β space of graph-directed self-similar sets in complete metric spaces.In Chapter Two, we are devoted to study in the dimension theory of dynamical systems, the multifractal analysis of equilibrium measures supported on conformal re-pellers and conformal Axiom A Diffeomorphisms. We prove the relationship between Hausdorff dimension and (?) di-mension,center Hausdorff measure and center (?) measure of level sets.Thus,This part extends the result of Yakov B.Pesin,H.Weiss(1997).