The Fractal Transformation Of Iterated Function System

Iteration function system is an important tool of studying fractal theory. In our thesis, we focus on two iterated function systems:hyperbolic ones and ones with Markov structure. We will study different ways of constructing codes which are essential of drawing fractal pictures. Among that, we will study the relation between them.We first studies the hyperbolic iterated function system. Through changing the norms, we make hyperbolic ones into one-step contracting ones which are much more easy to analysis.Then, given two iterated function systems F={f1,…,fN} and g= {g1,…,gN}, we define the attractors AF and Ag. Now we can define the fractal trans-formation f:AFâ†'Ag and the corresponding address structures CF and Cg. Through studying the address structures, we could obtain the following properties:If CF-< Cg, then the fractal transformation f is continuous; if CF=Cg, then the fractal transformation f is homeomorphic. A direct corollary is, if CF=Cg, then the corresponding dynamical systems TF and Tg are topologically conjugate.Next, we study one particular case of hyperbolic iterated function system:the one with Markov structure. Markov structure makes selecting codes much more easy, since symbolic code plays such important role of drawing the fractal pictures image. We give a detailed description on this conclusion.The last part are the applications. These include:drawing typical fractal pictures; analysis the mutual relation between fractal pictures; drawing continuous changing of fractal pictures.