Fractals In Iterated Function Systems

Iterated function system is an important branch of fractal geometry.This dissertation mainly discusses some fractal problems in iterated function systems:the best approximation of orbits in iterated function systems and the growth speed of the digits in Gauss-like iterated function systems.The whole dissertation is divided into five chapters.In the first chapter,we introduce the related backgrounds.In the second chapter,we mainly present the definitions and properties of conformal iterated function systems,as well as the preliminaries.In the third chapter,we consider the best approximation of orbits in iterated function systems.More precisely,for any x,y? J(attractor),in order to measure the approximation of the orbits of x and y,we define the shortest distance function Mn(x,y)as Mn(x,y)=max {k?N:?i-1(x)=?i+1(y),…,?i-k(x)=?i+k(y)for some 0 ?i?n-k which counts the run length of the longest same block among the first n digits of(x,y).In this chapter,we study the quantity Mn(x,y)in terms of the Renyi entropy,and we prove that for almost all(x,y)?J × J,Mn(x,y)increases at a log rate.The size of the set of points which satisfy Mn(x,y)increases at a prescribed rate are determined completely in conformal iterated function systems.In the fourth chapter,we consider the growth speed of the digits in Gauss-like iterated function systems.Let ?:N?R+ be an arbitrary positive function,define E(?)={x?J:?n(x)??(n)for infinitely many n? N?.We characterize the metric properties of E(?)in this chapter.We obtain a Borel-Bernstein type result on the Lebesgue measure of E(?),and determine the Hausdorff dimension of the set.In the final chapter,we review the main results and present some applications.Besides,we propose some questions for further research.