Some Problems On Fractal Geometry And Dynamical Systems

This thesis contains two main topi7cs related to fractal geometry and dynamical systems. The first two chapters are about iterated function systems and the third one is about the topological entropy of a certain dynamical system.In chapter one, we develop a theory for Hausdorff dimension and measure of self-conformal sets in complete metric spaces. This part is a try since G. A. Edgar and J. Golds gave the definition of conformal maps in complete metric spaces in 1999. It also extends the result of A. Schief(1996), which considered self-similar sets in complete metric spaces.Chapter two concerns graph-directed iterated function systems in Rd. It consists of two results. One is the equivalence of the OSC(open set condition), the SOSC(strong open set condition), and positivity of Hausdorff measure of the graph directed self-conformal sets in their dimension(when the system is made up of conformal maps, the attractor is called a self-conformal set). The other is a measure dimension estimate for graph-directed iterated function systems when they satisfy the double Lipschitz condition and the SOSC. We obtain the lower and upper bound estimate for the Hausdorff dimension of a list of useful measures.In chapter three, we study the topological entropy of the set of divergence points. Suppose is a transformation of a compact metric space (X, d) satisfying thespecification. For n N. we let Lnx = , where X denotes the Dirac measure atx. Let M(X) denote the family of probability measures on X, Y be a vector space withlinearly compatible metric and : M(X) Y be continuous and affine with respect to the weak topology on M(X). The set of divergence points is defined as following:We obtain that either all has the same limiting point or the topological entropy of the divergence points is as big as the whole space X. We also study the topological entropy of sup sets. i.e..where C is a closed convex subset of Y. Let M(f, X) denote the set of the f invariant probability measures on X, we havewhere h?(f) denotes the measure topological entropy of with respect to f.